How Technology Is Constantly Evolving And We Are Too With It

Technology is constantly evolving and we are too with it. We incorporate technology into our everyday lives to make them easier, provide entertain or be able to connect with one another. Arguably, no creation has made this more possible than Cyberspace. Prior to the twenty first century cyberspace was thought to be in the realm of science fiction, until a couple decades later technological breakthroughs brought it to existence. From that point on it has become essential to personal and global infrastructure. Storing our identity, personal information, wealth and for some our whole lives. Which makes it dangerous as well if abused. Cyberspace is a new frontier and we still don’t completely understand it. So how do we prosecute those who do harm to one another by means of cyberspace and cybercrime? Generally cybercrime branches of in to three major groups. Against a person, property or the government. A cybercrime against a person includes harassment and stalking a person through the internet/cyberspace. This is possible through the aids of email, social media and other sites or similar sorts. The types of harassment possible through cyberspace is sexual, racial, religious and others. Cybercrime against someone’s property include transmission of harmful programs and viruses. Also hacking corporate databases to steal information and money and email scams used to trick people to transfer money are all forms of cybercrime against property. The last and third major form ofShow MoreRelatedCause/Effect How Technology Influences Personal Relationships1095 Words   |  5 Pagesimpacts of digital technology on society. Technology has been evolving for hundreds of years. As it has become more advanced, the more it has taken a hold of the community. Digital technology is universal and there is no way to avoid it, but people need to start using it more responsibly. Everything gets taken for granted now that there are so many technological sources. Although there are a moderate amount of positive effects, the negative aspects outweigh them substantially. Technology use affects theRead MoreCan Technology Affect Your Mind?1607 Words   |  7 Pages1105 Emily Gilliam February 19, 2011 Can Technology affect your mind? iPod+ iPhone+ iPad= iBroke†Ã¢â‚¬ ¦and dysfunctional† Has your life begun to revolve around your cell phone? Are you checking your email, texting or tweeting more often than you speak to an actual human? Are you making life and death decisions at the computer. â€Å"Should I buy new iTunes or have gas for the rest of the week?† This is a growing problem. People have begun to let technology control their lives. They don’t have theRead MoreEssay on Nursing Informatics And Nursing889 Words   |  4 Pagesseen it in action many times. Are we as nurses changing with the times? What is nursing informatics? Why is it important to me? How do I rate on the nursing informatics knowledge scale? What is my plan to increase my knowledge base? These questions should be at the forefront of every nurse’s thoughts. The information age has come crashing down on us from every possible angle in our lives, it affects how we communicate, how we educate, how we socialize and how we raise our children. Thus, why wouldRead MorePositiv e And Negative Aspects Of Advertising1695 Words   |  7 Pagesconstant give and take from consumers. Furthermore, I have investigated certain technological benefits associated with advertising that improve effectiveness, and how advertising benefits consumers. On the contrary, I have evaluated some of the associated ethical issues, the promotion of unnecessary consumerism, psychological effects on learning, how younger generations are becoming less receptive, and the adverse affects on society as a whole. Regardless of the conflicting viewpoints it is best to be knowledgeableRead MoreAutomation In The Workforce. The Advancement Of Automation1435 Words   |  6 Pagesworkforce The advancement of automation has affected our everyday lives since the industrial revolution. Over the years we saw a drastic increase in unemployment due to the fact that machines and robots can now do the job more efficiently. Today we have adopted the idea of automation that we sometimes are unaware of the subtlety. We only become aware of the change when old technology advances or when there is a system malfunction while using the product. This monopoly is spreading through the workforceRead MoreGlobalization And The Global World1669 Words   |  7 PagesGlobalization is something that we see happening all around us. When we walk down the street it is impossible not to have an advertisement for major corporations like Nike, McDonalds, or Coca Cola flashing in front of your face. Everywhere our head turns, commercialism is there. It is unavoidable. The world around us is constantly developing, and that development begins with us. As the world keeps evolving, globalization is co nsidered inevitable. As humans, we naturally strive for success, seekingRead MoreMass Media Essay739 Words   |  3 Pagesduring the last century have literally changed our world and the way we get our information. These developments range from the telegraph, telephone, radio, television, and the Internet. All of which played a major role in not the way we receive information but also communicate. Instead of waiting days to get messages, we can now get them as they are happening. The following will discuss how each one has helped change the way that we receive our media and benefitted the American culture. The telegraphRead MorePrivacy Lost By David Holtzman1620 Words   |  7 Pagesnot be violated, and no Warrants shall issue, but upon probable cause, supported by Oath or affirmation, and particularly describing the place to be searched, and the persons or things to be seized (US 1). It is important to understand the protection we do have under this law- even if it is very limited. This law was created to ensure that the government has limitations on its powers, and that it cannot gather any information from people without first asking the court for a warrant. Does this meanRead MoreA Battle Between Minds1064 Words   |  4 Pagesâ€Å"Right† and â€Å"wrong† are such ambiguous terms and can only be personally defined by an individual’s beliefs and values. It is said that our values are defined predominately by our upbringing, but what if it is more neurologically ingrained than we had perceived? Doctor Roger Wolcott Sperry, neurophysiologist, won the Nobel Pri ze in 1981 for his discoveries concerning the functional specialization of the cerebral hemispheres, in which he studied the cognitive effects caused by severing the longitudinalRead MorePrivacy Lost By David Holtzman1386 Words   |  6 Pagesnot be violated, and no Warrants shall issue, but upon probable cause, supported by Oath or affirmation, and particularly describing the place to be searched, and the persons or things to be seized (US 1). It is important to understand the protection we do have under this law-even if it is very limited. This law was created to insure its citizens that the government has limitations on its powers, and that it cannot gather any information from people without first asking the court for a warrant. Does

Germany s Culture And Heritage - 1507 Words

Germany is a country in Western Europe, they are also wealthiest and most populated country in entire continent. Germany has the large amount of people in western Europe and also has the best economy system there. Germany’s landscapes are very similar to the United States, in Northern Germany that’s where you would find snowy mountains Mid-western would consist of the forested hills of the urbanized west. Even if Germany has a long history with everything from World War 2 and a lot of other events, the country is the youngest and newest even younger the United States. Germany. Before 1871 when Germany was called Prussia the land before was made up of many little kingdoms and Monarchies. This is why the Germans many consider themselves†¦show more content†¦Germany stayed but changed their name to Deutsches Reich which they had until the end of World War 2. In the 20th Century in Eastern Europe Germany was the most successful and powerful out of the European countries including Russia. They ended up crushing the Russians in the battle of Tannenberg. Russia was completely destroyed in all this and was forced to sign a peace treaty in March of 1918. Also during World War one the Germans made a ship that can be uses under water to surprise allies’ ships during the war. These boats where dubbed as German U boats by the allies, and are also known now as submarines today. In April of 1917 there was a cruise ship called the Lusitania which got shot up and sunk by a German -U-Boat forcing the United States of America into war against them. In September of 1919, this Austrian man by the name Adolf Hitler joined the politics. Adolf Hitler convinced people on the German committee that Germany was stabbed in the back by the Allies powers and the committee believed him and listened. They all came to the agreement that Germany was the best and was better than any country in the world and wanted all the Germans to live together to form a better, a greater Germany. The arty was racist and anti-Semitic which was understandable for the time being. In 1920 the small positions group was renamed the Nazi Party.Show MoreRelatedAdvantages And Disadvantages Of Tourism1661 Words   |  7 Pagesmoments among the time such as the beginning of the country in the medieval conquered by Romans, Mongolian and Chinese empires. In the 18’s century was influenced by the Latin countries as Italy, Spain and Portugal keeping even today the official language, the Latin. After the 19th century was overtook by Britain and Mussolini Italy follo wing a period of Germany domination in the time of the Second Great World War. The country won her independency in 1949 keeping strong relationship with the SovietRead MoreThe Importance of Archaeology1232 Words   |  5 Pagesis the study of historical and prehistorcial civilizations through the recovery and analysis of their materials culture. Moreover, it contains the study of human activity in the past. 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To attemptRead MoreUnesco And The World Heritage List1539 Words   |  7 PagesUNESCO and the World Heritage List After the devastation created by World War I, and II, UNESCO (United Nations Educational, Scientific, and Cultural Organization) was founded in 1945. This organization was created as a way to establish peace, based on humanity’s intellectual and moral solidarity. Then in 1972, UNESCO founded the World Heritage Convention as a way to protect sites of exceptional worldwide importance (UNESCO, 2012). This convention was later ratified by 191 countries, making itRead MoreReligious Transformations Of The United States1275 Words   |  6 Pagesreligious transformations. The following paragraphs will briefly examine five different transformations that have occurred during the history of religion in the United States. Pluralism is by definition, the coexistence of multiple, groups, religions, cultures, etc. One example of Pluralism that sticks out is the history of colonial Protestantism. As discussed in class, Protestantism stemmed from desire to reform the Roman Catholic Church. The actions of three figures, Martin Luther, John Calvin, and KingRead MoreCultural Background : A Cultural Perspective As A Future Counselor808 Words   |  4 Pagesfreckles definitely do not go unnoticed. â€Å"It’s the Irish in me,† I often respond, but if a person could look beyond my spotted skin to within, they’d see that I identify most with my German heritage. When my paternal grandparents immigrated from Germany to the United States in the 1950’s, they brought their culture with them, influencing future generations, and especially myself. After analyzing my three family generations, I was proud to learn how my cultural background has greatly contributed to my

Total Quality Management Proposal for Smart Pack Ltd.

Question: Discuss about the Total Quality Management Proposal for Smart Pack Ltd. Answer: Introduction Lego is a line of Plastic Construction toys and its flagship product, Lego, consists of colorful interlocking plastic bricks which could be connected and assembled in myriads of ways. The company has been using a strong, resilient material, ABS (Acrylonitrile butadiene styrene, since 1963 to manufacture its building pieces. In order to expand its business in Asian markets, the firm is looking for a subcontractor in Australia and for this it has approached a small niche manufacturer, Smart Pack Ltd. in Australia. The new manufacturer is expected to assemble small LEGO sets and manufacture toys that are highly compatible to Australian markets. But, prior to this the company needs to conduct a Total Quality Management process to be the most deserving subcontractor of Lego (ASQ Team, 2016). TQM basically describes the management approach to attain a long-term success by responding to the customer needs readily and satisfying them in every sense. The TQM approach asks all the participating employees of the company to work in the direction of improvement in the firms operations and introducing innovative ideas for reforming the workplace culture. A permanent climate is introduced under this process in which all the members of a company continuously improve their skills and capabilities to deliver the best services and high Quality products to its customers. There is a keen requirement of this assessment because Lego is a leading giant in the Toy industry and has been widely accepted over all the foreign markets just because of its commitment towards the quality of the products. The following Business report has been precisely written to highlight the core principles and techniques of the TQM process that are to be undertaken to produce the best outcomes to the parent company (Wiley, 2012). The report also deals with the precise implementation of the TQM approaches within the firm and for this even an action plan has been generated. The prime aim of this report is to convince the owner about the abilities of the Smart Pack Ltd. and why it should be readily considered as a sub-contractor in Australia over the other competitors in the markets. Core Principles of TQM In order to involve a great level of customer satisfaction, an organization needs to follow certain core principles: The organization should follow approaches and develop operations that are entirely customer focused and tends to deliver ultimate satisfaction to them. The tasks being done in order to improve the quality of services are only completed if they are readily accepted by the customers (Baumeyer, 2015). The employees should be promoted to be greatly involved in all the crucial decisions of the company. For this purpose, all types of restrictions should be removed from the workplace and the employees should be empowered with the power of decision making. Process centered thinking should become the utmost priority of the organization as the organization which considers efficiency of the processes over the outcomes is the ones that tend to be most successful (Hashmi, 2015). The Integrated system also forms a unique principle of the TQM. In integrated system all the participants knows about the vision and mission of the company and hence align all the perceptions and activities to achieve them. The plans that begin with a precise strategy have a high success rate. The Strategic and Systematic approach towards the attainment of goals gives a sound platform of effective service delivery to the customers. The approaches after implementation need a continuous improvement and adoption of new trends. The creative and innovative thinking has a capability to make the future decisions more efficient and effective. Decision making is an integral process in any organization but at the same time it is the responsibility of the executives to make this process fact-based instead of an abstract one. The fact-based decisions often allow the organization members to consider those decisions that are beneficial for all at the same time. Hence, with such an approach, the decision making process becomes an unbiased one (Scheid, 2010). Communication is the last but the most important principle of the TQM process. Effective communication tools allow the exact ideas and goals to be promoted among the participants of the organization. This also allows the executives to gain feedbacks of the subordinates over certain decisions and even increase their participation in the crucial aspects of the firm. Required Tools and Techniques There are a number of tools and techniques which are to be installed in order to achieve success while implementing the process elements of TQM. The Qualitative tools consist of subjective inputs which do not measure any numerical value, but give an abstract idea of something immeasurable. The other types of tools are quantitative ones which involve the analysis of the objective data and only consider something with which numerical value is attached (Presentationeze Team, 2016). The best known tools and techniques of a TQM include: The Flowcharts that could be efficiently used in examining activities or in brainstorming To identify the Cause and Effect processes, the researchers could implement the Fishbone or Ishikawa Diagrams. The primary categories are generally pre-determined in such cases (Jafari Setak, 2010). As the owner has already given a deadline of the submission of the business report to the subcontractor, hence, there could also be an implementation of Run Charts in which changes and data are plotted against timeline. As the TQM process highly focuses on improvement of the quality of services delivered to the customers, so a number of process improvement tools such as FMEA (Failure Mode Effects Analysis), PDCA (Plan Do Check Act), SIPOC Analysis and Statistical Control tools could also be included (Singh Grover, 2012). To efficiently implement and maintain the product and service quality, Smart Pack Ltd., being a small firm could also direct its resources in implementing the best TQM techniques of all time like Brainstorm Analysis, Fault Tree Analysis and Hazop Analysis (Gomes, 2011). Implementation of the TQM As Smart Pack Ltd. has to demonstrate its capabilities and abilities to Lego in order to be chosen as the best subcontractor in Australia above all the others, hence, it becomes its prime responsibility to give a candid description of what all will be included in the TQM process being carried out in the company. TQM approach demands a firm to deliver the best quality of goods and services to the customers and even satiate the needs of the workforce. TQM entirely talks about Quality and its improvement in the coming future and this definitely needs a precise plan that is to be communicated and followed by all the participants of the organization. As the company, Smart Pack Ltd. is planning to prove its capabilities in front of the owner as a potential subcontractor in Australia, so it needs to follow a certain set of steps that depicts the actual implementation of the TQM process in the entire organization. These steps are: Clarification of Mission, Vision and Values: The employees participating in the TQM process need to know where their talents and skills are being used and what exactly the vision of the firm is. When they are asked to deliver services that are goal-oriented, then their skills are utilized in a better manner. Orientation of the newly recruited employees is necessary to communicate the true values of the firm and its candid priorities while executing its operations. The Identification of CSF: CSF or the Critical Success Factors are to be identified in the initial stages of implementation of the TQM processes. This identification helps the executives to focus on those things that demands optimum resources, yet promises a huge success of the firm. Moreover, these performance-based factors also help in determining how well a company is performing and what all reformations are needed for the same. Identification of the Key Target Audiences and Customer Groups: Identification of the target groups is crucial prior to the development of products and services. The actual needs and demands of the customers help in framing customized services that have the potential in satiating all sorts of tastes and perceptions in the markets. The Key customer groups in this process include the Employees, the Customers, Vendors, Suppliers and other types of Volunteers that are highly influenced by the outcomes of the TQM process (Sharma, et al., 2014). Gathering Customer and Employee feedbacks: To evaluate the performance of the firm, the executives need to gather feedbacks from myriads of stakeholders. In the TQM processes these feedbacks could be gathered from the Customers and Employees. This will definitely increase their participation in the crucial decision making processes for the organization. Developing and Implementing Survey Tools: The Survey tools often help in identifying that what is best for the customers and in what manner, the organization could meet these requirements. The surveys often provide a starting point for the improvements and also demonstrate progress as improvement plans are implemented (Chandra, 2013). Planning and Implementing Improvement Plan: The customer feedbacks and survey tools help in gathering feedbacks over the current operations of the firm. After the baseline is established, the executives could develop and implement Improvement plans based on these feedbacks The Improvement Plans should have candid operations and should be accepted by all the participants of the organization. Monitoring the TQM process: Monitoring is important to ensure that there is consistent progress towards the goals. This process also helps the executives to determine what all resources have been used till now and what all are to be added for the smooth functioning of the company. Monitoring also helps in evaluating the change in priorities and objectives with the course of time to adapt to the ongoing trends in the business world. Considering the Technological Reformations: The technological elements are reforming at a rapid speed and the organization needs to keep pace with it. The TQM process could only be implemented in an effective manner if the executives consider all the leading changes at the technological level and implement those that could foster the success rates in the organization. Incorporate the achievements and Satisfaction data in the marketing plan: The Satisfaction experienced by the customers by the quality of the product and services of the firm could be used as a marketing tool to attract other customers in myriads of new markets. Similarly, the achievements of the company could also be showcased to gain the loyalty of the customers. Action Plan for the Implementation Below is the Action Plan for two years of implementation of the TQM process in Smart Pack Ltd. Tasks Strategies Person (s) responsible Resources needed Timeline 1. Communication of the exact mission, vision and values to all the stakeholders - Arrangement of weekly and monthly meetings - Effective Training and Interaction sessions - Leaders of the HR department - Experts hired from other organizations - Senior and middle managers - Efficient Conference rooms - Separate budget for meetings - Efficient network systems for upgraded online platforms (Lakhe Mohanty, 2014). - Within 1-2 months of the commencement of the TQM process. 2. Identification of the Critical Success Factors (CSF) - Considering the previous TQM activities occurred in the organization - Watching closely the trends during the success and downfall of the company - Financial manager with its team - Regulatory team with leaders from all the subunits of the organization. - Previous Annual reports and budgets - Reports on availability and lacking of resources within the company. - First Quarter of the first year of Implementation 3. Identification of Key Target Audiences and Customer groups - Launching sample products in the market to record the response and identify target consumers - Taking help from the local marketers to analyze the perceptions of consumers from different markets. - HR manager with the team - Leaders with efficient communication skills - Sample products - Local marketers reports - First two Quarters of the first year of implementation 4. Gathering Feedbacks - Surveys and Personal interviews - Asking customers to fill up the feedback forms. - Employees with interactive power - Company representatives to conduct surveys - Effective Survey Questions - Feedback forms and systems to process them (Floss, et al., 2012). The last quarter of the first year of implementation 5. Developing and Implementing Survey tools - Evaluating the texts for the best survey tools - Hiring experts from another firm to create effective survey questions - Leaders with good critically evaluation power - Representatives to conduct periodic checks and evaluation - The audit reports of the previous years - List of available survey tools in the company - Should begin from the third quarter of the first year of implementation 6. Planning and Implementing Improvement Plan - Design the layout of the best improvement plan - Compare the existing plans with that of the other leaders in the market - The team planning for improvements - Representatives of the company with intellect and innovation - Previous improvement plan - List of ongoing trends in the markets - A List of the improvements adopted by other companies - First quarter of the second year of implementation 7. Monitoring the TQM process -Regular Audits - Meetings - The Audit team - Leaders from all the subunits - The elements that would be undertaken under evaluation - Separate budget for hiring the monitoring team (Caro, 1983). - During the third and fourth quarter of the second year of implementation 8. Considering the Technological Reformations - Observing the current and future technological trends - The experts from the IT sector - Representatives with a command in IT field - Latest Technological tools and techniques - Throughout both the implementation year 9. Incorporation of the Achievements and Satisfaction data in the marketing plan - Recording periodically the success and satisfaction data - Leaders with critical evaluation abilities - Audit and Monitoring Reports - The last two quarters of the second year of implementation Recommendations and Conclusion The TQM process promises a great improvement in a firm and prepares it for a better future. To be the most desirable subcontractor, Smart Pack Ltd. should efficiently follow the above mentioned implementation steps in a given time frame and should even monitor the progress periodically. Moreover, as Lego is highly committed towards the quality of its products, hence, its subcontractor also has to consider the quality of the products and services above all the other operations in the company. References ASQ Team, 2016. What is Total Quality Management (TQM)?. What is Total Quality Management (TQM)?. Baumeyer, K., 2015. Five Principles of Total Quality Management (TQM). Five Principles of Total Quality Management (TQM). Caro, J. C., 1983. Planning and Implementing Total Quality Management (TQX) in a Naval service organization: A Case Study of Fleet Numerical oceanography Center: DTIC, Available at: https://www.dtic.mil/dtic/tr/fulltext/u2/a242311.pdf Chandra, P., 2013. A Study on Implementation of Total Quality Management in Businesses. International Journal of Engineering Science and Innovative Technology (IJESIT), May, 2(3), pp. 1-7. Floss, G., Lynch, T. Naughton, J., 2012. Total Quality Management Master Plan: GOAL/QPC Research Committee, Available at: https://www.goalqpc.com/cms/docs/tqmMasterPlan.pdf Gomes, S., 2011. Different techniques for TQM: Benchmarking: Wordpress, Available at: https://xisspm.files.wordpress.com/2011/07/chap-4-tqm-techniques.pdf Hashmi, K., 2015. Introduction and Implementation of Total Quality Management (TQM). Introduction and Implementation of Total Quality Management (TQM). Jafari, S. Setak, M., 2010. Total Quality Management Tools and Techniques: The Quest for an Implementation Roadmap. Total Quality Management Tools and Techniques: The Quest for an Implementation Roadmap. Lakhe, R. Mohanty, R., 2014. Total Quality Management Concepts, Evolution and Acceptability in Developing Economies, Available at: https://www.icesi.edu.co/blogs/bitacoraestrategia0314/files/2014/03/21-Total-Quality-Management-concepts.pdf Presentationeze Team, 2016. TQM Tools and Techniques. [Online] Available at: https://www.presentationeze.com/presentations/tqm-tools-and-techniques/ Scheid, J., 2010. Analysis of TQM Quality Concepts. Analysis of TQM Quality Concepts, 16 December. Sharma, S., Gupta, S. Singh, R., 2014. Implementation Of TQM For Improving Organizational Effectiveness. International Journal of Application or Innovation in Engineering Management, September.3(9). Singh, M. Grover, S., 2012. Tools and techniques for quality management in manufacturing industries: YMCA University of Science Technology, Available at: https://ymcaust.ac.in/tame2012/cd/industrial/IE-30.pdf Wiley, 2012. Total Quality Management. In: Total Quality Management, pp. 1-35, Available at: https://www.wiley.com/college/sc/reid/chap5.pdf

Matrices in Matlab Essay Example

Matrices in Matlab Paper Matrices in Matlab You can think of a matrix as being made up of 1 or more row vectors of equal length. Equivalently, you can think of a matrix of being made up of 1 or more column vectors of equal length. Consider, for example, the matrix ? ? 1 2 3 0 A = ? 5 ? 1 0 0 ? . 3 ? 2 5 0 One could say that the matrix A is made up of 3 rows of length 4. Equivalently, one could say that matrix A is made up of 4 columns of length 3. In either model, we have 3 rows and 4 columns. We will say that the dimensions of the matrix are 3-by-4, sometimes written 3 ? . We already know how to enter a matrix in Matlab: delimit each item in a row with a space or comma, and start a new row by ending a row with a semicolon. gt;gt; A=[1 2 3 0;5 -1 0 0;3 -2 5 0] A = 1 2 3 0 5 -1 0 0 3 -2 5 0 We can use Matlab’s size command to determine the dimensions of any matrix. gt;gt; size(A) ans = 3 4 That’s 3 rows and 4 columns! Indexing Indexing matrices in Matlab is similar to the indexing we saw with ve ctors. The di? erence is that there is another dimension 2. To access the element in row 2 column 3 of matrix A, enter this command. 1 2Copyrighted material. See: http://msenux. redwoods. edu/Math4Textbook/ We’ll see later that we can have more than two dimensions. 76 Chapter 2 Vectors and Matrices in Matlab gt;gt; A(2,3) ans = 0 This is indeed the element in row 2, column 3 of matrix A. You can access an entire row with Matlab’s colon operator. The command A(2,:) essentially means â€Å"row 2 every column† of matrix A. gt;gt; A(2,:) ans = 5 -1 0 0 Note that this is the second row of matrix A. Similarly, you can access any column of matrix A. The notation A(:,2) is pronounced â€Å"every row column 2† of matrix A. gt;gt; A(:,2) ans = 2 -1 -2 Note that this is the second column of matrix A. You can also extract a submatrix from the matrix A with indexing. Suppose, for example, that you would like to extract a submatrix using rows 1 and 3 and columns 2 and 4. gt;gt; A([1,3],[2,4]) ans = 2 0 -2 0 Study this carefully and determine if we’ve truly selected rows 1 and 3 and columns 2 and 4 of matrix A. It might help to repeat the contents of matrix A. Section 2. 2 Matrices in Matlab 77 gt;gt; A A = 1 5 3 2 -1 -2 3 0 5 0 0 0 You can assign a new value to an entry of matrix A. gt;gt; A(3,4)=12 A = 1 2 5 -1 3 -2 3 0 5 0 0 12 When you assign to a row, column, or submatrix of matrix A, you must replace the contents with a row, column, or submatrix of equal dimension. For example, this next command will assign new contents to the ? rst row of matrix A. gt;gt; A(1,:)=20:23 A = 20 21 22 5 -1 0 3 -2 5 23 0 12 There is an exception to this rule. If the right side contains a single number, then that number will be assigned to every entry of the submatrix on the left. For example, to make every entry in column 2 of matrix A equal to 11, try the following code. gt;gt; A(:,2)=11 A = 20 11 5 11 3 11 22 0 5 23 0 12 It’s interesting what hap pens (and very powerful) when you try to assign a value to an entry that has a row or column index larger than the corresponding dimension of the matrix. For example, try this command. 78 Chapter 2 Vectors and Matrices in Matlab gt;gt; A(5,5)=777 A = 20 11 5 11 3 11 0 0 0 0 22 0 5 0 0 23 0 12 0 0 0 0 0 0 777 Note that Matlab happily assigns 777 to row 5, column 5, expanding the dimensions of the matrix and padding the missing entries with zeros. gt;gt; size(A) ans = 5 5 The Transpose of a MatrixYou can take the transpose of a matrix in exactly the same way that you took the transpose of a row or column vector. For example, form a â€Å"magic† matrix with the following command. gt;gt; A=magic(4) A = 16 2 5 11 9 7 4 14 3 10 6 15 13 8 12 1 You can compute AT with the following command. gt;gt; A. ’ ans = 16 2 3 13 5 11 10 8 9 7 6 12 4 14 15 1 Section 2. 2 Matrices in Matlab 79 Note that the ? rst row of matrix AT was previously the ? rst column of matrix A. The second row of matrix AT was previously the second column of matrix A, and so on for the third and fourth columns of matrix AT . In essence, taking the transpose re? cts the matrix A across its main diagonal (upper left corner to lower right corner), so the rows of A become columns of AT and the columns of A become rows of AT . Building Matrices Matlab has some powerful capabilities for building new matrices out of one or more matrices and/or vectors. For example, start by building a 2 ? 3 matrix of ones. gt;gt; A=ones(2,3) A = 1 1 1 1 1 1 Now, build a new matrix with A as the ? rst column and A as the second column. As we are not starting a new row, we can use either space or commas to delimit the row entries. gt;gt; C=[A A] C = 1 1 1 1 1 1 1 1 1 1 1 1On the other hand, suppose that we want to build a new matrix with A as the ? rst row and A as the second row. To start a new row we must end the ? rst row with a semicolon. gt;gt; C=[A; A] C = 1 1 1 1 1 1 1 1 1 1 1 1 Let’s create a 2 ? 3 matrix of all zeros. 80 Chapter 2 Vectors and Matrices in Matlab gt;gt; D=zeros(2,3) D = 0 0 0 0 0 0 Now, let’s build a matrix out of the matrices A and D. gt;gt; E=[A D;D A] E = 1 1 1 1 0 0 0 0 1 1 0 0 0 0 1 1 0 0 1 1 0 0 1 1 The possibilities are endless, with one caveat. The dimensions must be correct or Matlab will report an error. For example, create a 2 ? 2 matrix of ones. gt;gt; A=ones(2,2) A = 1 1 1 1 And a 2 ? 3 matrix of zeros. gt;gt; B=zeros(2,3) B = 0 0 0 0 0 0 It’s possible to build a new matrix with A and B as row elements. gt;gt; C=[A B] C = 1 1 1 1 0 0 0 0 0 0 Section 2. 2 Matrices in Matlab 81 But it’s not possible to build a new matrix with A and B as column elements. gt;gt; C=[A;B] Error using ==gt; vertcat CAT arguments dimensions are not consistent. This happens because A has 2 columns, but B has 3 columns, so the columns don’t line up. We’ll see in later work that the matrix building capabilities of Matlab are a powerful ally . Scalar-Matrix MultiplicationIf asked to multiply a matrix by a scalar, one would hope that the operation of scalar-matrix multiplication would be carried out in exactly the same manner as scalar-vector multiplication. That is, simply multiply each entry of the matrix by the scalar. Example 1. If A is the matrix ? 1 2 3 A = 3? 4 5 6? , 7 8 9 ? perform the scalar-matrix multiplication 3A. Simply multiply 3 times each ? 1 3A = 3 ? 4 7 entry of the matrix. ? ? ? 2 3 3 6 9 5 6 ? = ? 12 15 18 ? 8 9 21 24 27 Matlab understands scalar-matrix multiplication. First, enter matrix A. gt;gt; A=[1 2 3;4 5 6;7 8 9] A = 1 2 3 4 5 6 7 8 9 Now compute 3A. 82 Chapter 2Vectors and Matrices in Matlab gt;gt; 3*A ans = 3 12 21 6 15 24 9 18 27 Matrix Addition If two matrices have the same dimension, then add the matrices by adding the corresponding entries in each matrix. Example 2. If A and B are the matrices ? ? ? ? 1 1 1 1 1 1 A = ? 2 2 2? and B = ? 1 1 1? , 3 3 3 1 1 1 ? nd the sum A + B. Simply add the corresponding entries. ? ? ? ? ? ? 1 1 1 1 1 1 2 2 2 A + B = ? 2 2 2? + ? 1 1 1? = ? 3 3 3?. 3 3 3 1 1 1 4 4 4 Matlab understands matrix addition. gt;gt; A=[1 1 1;2 2 2;3 3 3]; B=[1 1 1;1 1 1;1 1 1]; gt;gt; A+B ans = 2 2 2 3 3 3 4 4 4 This is identical to the hand-calculated sum above.Let’s look what happens when the dimensions are not the same. Example 3. If A and B are the matrices Section 2. 2 ? 1 1 1 A = ? 2 2 2? 3 3 3 ? then ? nd the sum A + B. Note the dimensions of each matrix. Matrices in Matlab 83 and B= 1 1 1 , 1 1 1 gt;gt; A=[1 1 1;2 2 2;3 3 3]; B=[1 1 1;1 1 1]; gt;gt; size(A) ans = 3 3 gt;gt; size(B) ans = 2 3 The matrices A and B do not have the same dimensions. Therfore, it is not possible to sum the two matrices. gt;gt; A+B Error using ==gt; plus Matrix dimensions must agree. This error message is completely expected. One ? nal example is in order. Example 4. If matrix A is ? 1 1 1 A = ? 2 2 2? 3 3 3 compute A + 1. Note that this addition of a matrix and a scalar makes no sense. ? ? 1 1 1 A + 1 = ? 2 2 2? + 1 3 3 3 ? 84 Chapter 2 Vectors and Matrices in Matlab The dimensions are all wrong. However, this is such a common occurrence in algebraic calculations (as we will see throughout the course), Matlab allows this matrix-scalar addition. gt;gt; A=[1 1 1;2 2 2;3 3 3]; gt;gt; A+1 ans = 2 2 2 3 3 3 4 4 4 Matlab simply adds 1 to each entry of the matrix A. That is, Matlab interprets A + 1 as if it were the matrix addition of Example 2. Matrix addition enjoys several properties, which we will ask you to explore in the exercises. . Addition is commutative. That is, A + B = B + A for all matrices A and B having the same dimension. 2. Addition is associative. That is, (A + B) + C = A + (B + C), for all matrices A, B, and C having the same dimension. 3. The zero matrix is the additive identity. That is, if A is m ? n and 0 is an m ? n matrix of all zeros, then A + 0 = A. 4. Each matrix A has an additive inverse. Form the matrix ? A by negatin g each entry of the matrix A. Then, A + (? A) = 0. Matrix-Vector Multiplication Consider the linear system of three equations in three unknowns. 2x + 3y + 4z = 6 3x + 2y + 4z = 8 5x ? 3y + 8x = 1. 2. 1) Because each of the corresponding entries are equal, the following 3 ? 1 vectors are also equal. ? ? ? ? 2x + 3y + 4z 6 ? 3x + 2y + 4z ? = ? 8 ? 5x ? 3y + 8x 1 Section 2. 2 Matrices in Matlab 85 The left-hand vector can be written as a vector sum. ? ? ? ? ? ? ? ? 2x 3y 4z 6 ? 3x ? + ? 2y ? + ? 4z ? = ? 8 ? 5x ? 3y 8z 1 Scalar multiplication can be used to factor the variable out of each vector on the left-hand side. ? ? ? ? ? ? ? ? 2 3 4 6 x? 3? + y? 2 ? + z? 4? = ? 8? (2. 2) 5 ? 3 8 1 The construct on the left-hand side of this result is so important that we will pause to make a de? nition.Definition 5. Let ? 1 , ? 2 , . . . , and ? n be scalars and let v1 , v2 , . . . , and vn be vectors. Then the construction ? 1 v1 + ? 2 v2 +  ·  ·  · + ? n vn is called a linear combination of the vectors v1 , v2 , . . . , and vn . The scalars ? 1 , ? 2 , . . . , and ? n are called the weights of the linear combination. For example, we say that ? ? ? ? ? ? 2 3 4 ? 3? + y? 2 ? + z? 4? x 5 ? 3 8 is a linear combination of the vectors [2, 3, 5]T , [3, 2, ? 3]T , and [4, 4, 8]T . 3 Finally, we take one last additional step and write the system (2. 2) in the form ? ? ? ? 2 3 4 x 6 ? 3 2 4 y ? = ? 8?. (2. 3) 5 ? 8 z 1 Note that the system (2. 3) has the form Ax = b, where 3 Here we use the transpose operator to save a bit of space in the document. 86 Chapter 2 ? Vectors and Matrices in Matlab ? ? x ? y ? , x= z ? ? 6 ? 8?. b= 1 ? 2 3 4 A = ? 3 2 4? , 5 ? 3 8 and The matrix A in (2. 3) is called the coe? cient matrix. If you compare the coe? cient matrix in (2. 3) with the original system (2. 1), you see that the entries of the coe? cient matrix are simply the coe? cients of x, y, and z in (2. 1). On right-hand side of system (2. 3), the vector b = [6, 8, 1]T contains the n umbers on the right-hand side of the original system (2. ). Thus, it is a simple matter to transform a system of equations into a matrix equation. However, it is even more important to compare the left-hand sides of system (2. 2) and system (2. 3), noting that ? ? ? ? ? ? ? ? 2 3 4 x 2 3 4 ? 3 2 4 y ? = x? 3? + y? 2 ? + z? 4?. 5 ? 3 8 z 5 ? 3 8 This tells us how to multiply a matrix and a vector. One takes a linear combination of the columns of the matrix, using the entries in the vector as weights for the linear combination. Let’s look at an example of matrix-vector multiplication Example 6. Multiply the matrix and vector ? ? 1 2 ? 3 1 ? 3 0 4 ? ? ? 2 ? . 0 ? 2 3 To perform the multiplication, take a linear combination of the columns of the matrix, using the entries in the vector as weights. ? ? ? ? ? ? ? ? 1 2 ? 3 1 1 2 ? 3 ? 3 0 4 ? ? ? 2 ? = 1 ? 3 ? ? 2 ? 0 ? + 3 ? 4 ? 0 ? 2 2 3 0 ? 2 2 ? ? ? ? ? ? 1 ? 4 ? 9 ? 3 ? + ? 0 ? + ? 12 ? = 0 4 6 ? ? ? 12 ? 15 ? = 10 Itâ€℠¢s important to note that this answer has the same number of entries as does each column of the matrix. Section 2. 2 Matrices in Matlab 87 Let’s see if Matlab understands this form of matrix-vector multiplication. First, load the matrix and the vector. gt;gt; A=[1 2 -3;3 0 4;0 -2 2]; x=[1; -2; 3]; Now perform the multiplication. gt;gt; A*x ans = -12 15 10 Note this is identical to our hand calculated result. Let’s look at another example. Example 7. Multiply Ax, where A= 1 1 1 2 0 ? 2 and x = 1 . 1 If you try to perform the matrix-vector by taking a linear combination using the entries of the vectors as weights, Ax = 1 1 1 2 0 ? 2 1 1 1 1 =1 +1 +? . 1 2 0 ? 2 (2. 4) The problem is clear. There are not enough weights in the vector to perform the linear combination. Let’s see if Matlab understands this â€Å"weighty† problem. gt;gt; A=[1 1 1;2 0 -2]; x=[1; 1]; gt;gt; A*x Error using ==gt; mtimes Inner matrix dimensions must agree.Inner dimensions? Let†™s see if we can intuit what that means. In our example, matrix A has dimensions 2 ? 3 and vector x has dimensions 2 ? 1. If we juxtapose these dimensions in the form (2? 3)(2? 1), then the inner dimensions don’t match. 88 Chapter 2 Vectors and Matrices in Matlab Dimension Requirement. If matrix A has dimensions m ? n and vector x has dimensions n ? 1, then we say â€Å"the innner dimensions match,† and the matrix-vector product Ax is possible. In words, the number of columns of matrix A must equal the number of rows of vector x. Matrix-Matrix Multiplication We would like to extend our de? ition of matrix-vector multiplication in order to ? nd the product of matrices. Here is the needed de? nition. Definition 8. Let A and B be matrices and let b1 , b2 , . . . , and bn represent the columns of matrix B. Then, AB = A b1 , b2 , . . . , bn = Ab1 , Ab2 , . . . , Abn . Thus, to take the product of matrices A and B, simply multiply matrix A times each vector column of matri x B. Let’s look at an example. Example 9. Multiply 1 2 3 4 1 ? 2 . 2 1 We multiply the ? rst matrix times each column of the second matrix, then use linear combinations to perform the matrix-vector multiplications. 1 2 3 4 1 ? = 2 1 = 1 = 1 2 3 4 1 , 2 1 2 3 4 ? 2 1 1 2 1 2 +2 , ? 2 +1 3 4 3 4 5 0 11 ? 2 Let’s see if Matlab understands this form of matrix-matrix multiplication. First, load the matrices A and B. gt;gt; A=[1 2;3 4]; B=[1 -2;2 1]; Now, multiply. Section 2. 2 Matrices in Matlab 89 gt;gt; A*B ans = 5 11 0 -2 Note that this result is indentical to our hand calculation. Again, the inner dimensions must match or the matrix-matrix multiplication is not possible. Let’s look at an example where things go wrong. Example 10. Multiply 1 1 1 2 0 ? 2 1 2 . 3 4 When we multiply the ? rst matrix times each column of the second matrix, we immediately see di? ulty with the dimensions. 1 1 1 2 0 ? 2 1 2 = 3 4 1 1 1 2 0 ? 2 1 , 3 1 1 1 2 0 ? 2 2 4 (2. 5) In the ? rst column of the matrix product, the matrix-vector multiplication is not possible. The number of columns of the matrix does not match the number of entries in the vector. Therefore, it is not possible to form the product of these two matrices. Let’s see if Matlab understands this dimension di? culty. gt;gt; A=[1 1 1;2 0 -2]; B=[1 2;3 4]; gt;gt; A*B Error using ==gt; mtimes Inner matrix dimensions must agree. The error message is precisely the one we would expect. Dimension Requirement.If matrix A has dimensions m ? n and matrix B has dimensions n ? p, then we say â€Å"the inner dimensions match,† and the matrix-matrix product AB is possible. In words, the number of columns of matrix A must equal the number of rows of matrix B. Let’s look at another example. 90 Chapter 2 Vectors and Matrices in Matlab Example 11. Multiply ? 1 2 1 1 1 ? AB = 1 ? 2 ? . 2 0 ? 2 2 0 Load the matrices A nd B into Matlab and check their dimensions. gt;gt; A=[1 1 1;2 0 -2]; B=[1 2;1 -2; 2 0]; gt;gt; size(A) ans = 2 3 gt;gt; size(B) ans = 3 2 Thus, matrix A has dimensions 2 ? 3 and B has dimensions 3 ? . Therefore, the inner dimensions match (they both equal 3) and it is possible to form the product of A and B. gt;gt; C=A*B C = 4 -2 ? 0 4 Note the dimensions of the answer. gt;gt; size(C) ans = 2 2 Recall that A was 2 ? 3 and B was 3 ? 2. Note that the â€Å"outer dimensions† are 2 ? 2, which give the dimensions of the product. Dimensions of the Product. If matrix A is m ? n and matrix B is n ? p, then the dimensions of AB will be m ? p. We say that the â€Å"outer dimensions give the dimension of the product. † Section 2. 2 Matrices in Matlab 91 Properties of Matrix MultiplicationMatrix multiplication is associative. That is, for any matrices A, B, and C, providing the dimensions are right, (AB)C = A(BC). Let’s look at an example. Example 12. Given A= 2 2 , 3 3 B= 1 1 , 2 5 and C = 3 3 , 2 5 use Matlab to demonstrate that (AB)C = A(BC). Load the matrices A, B, and C into Matlab, then calculate the left-hand side of (AB)C = A(BC). gt;gt; A=[2 2;3 3]; B=[1 1;2 5]; C=[3 3;2 5]; gt;gt; (A*B)*C ans = 42 78 63 117 Next, calculate the right-hand side of (AB)C = A(BC). gt;gt; A*(B*C) ans = 42 78 63 117 Hence, (AB)C = A(BC). Matrix Multiplication is Associative.In general, if A, B, and C have dimensions so that the multiplications are possible, matrix multiplication is associative. That is, it is always the case that (AB)C = A(BC). 92 Chapter 2 Vectors and Matrices in Matlab Unfortunately, matrix multiplication is not commutative. That is, even if A and B are of correct dimensions, it is possible that AB = BA. Let’s look at an example. Example 13. Let A= 1 2 3 4 and B = 3 5 . 2 7 Do the matrices A and B commute? That is, does AB = BA? Load the matrices into Matlab, then compute AB. gt;gt; A=[1 2;3 4]; B=[3 5;2 7]; gt;gt; A*B ans = 7 19 17 43 Now compute BA. gt;gt; B*A ans = 18 23 6 32 Thus, AB = BA. Matrix Multiplication is not Commutative. In general, even if the dimensions of A and B allow us to reverse the order of multiplication, matrices A and B will not commute. That is, AB = BA. Any change in the order of multiplication of matrices will probably change the answer. Some matrices do commute, making this even more complicated. Section 2. 2 Matrices in Matlab 93 gt;gt; A=[5 3;7 4],B=[-4 3;7 -5]; gt;gt; A*B ans = 1 0 0 1 gt;gt; B*A ans = 1 0 0 1 In this case, AB = BA. However, in general, matrix multiplication is not commutative. The loss of the commutative property is not to be taken lightly.Any time you change the order of multiplication, you are risking an incorrect answer. There are many insidious ways that changes of order can creep into our calculations. For example, if you multiply the left-hand side of equation on the left by a matrix A, then multiply the right-hand side of the equation on the right by the same matrix A, you’ve changed the order and should expect an incorrect answer. W e will explore how the loss of the commutative property can adversely a? ect other familiar algebraic properties in the exercises. Here is a list of matrix properties you can depend on working all of the time.Let A and B be matrices of the correct dimension so that the additions and multiplications that follow are possible. Let ? and ? be scalars. A(B + C) = AB + AC (A + B)C = AC + BC. (? + ? )A = ? A + ? A ? (A + B) = ? A + ? B. ?(? A) = ( )A. (? A)B = ?(AB) = A(? B). For example, as stated above, matrix multiplication is distributive over addition. That is, A(B + C) = AB + AC. gt;gt; A=[2 3;-1 4]; B=[1 2;0 9]; C=[-3 2;4 4]; gt;gt; A*(B+C) ans = 8 47 18 48 gt;gt; A*B+A*C ans = 8 47 18 48 94 Chapter 2 Vectors and Matrices in Matlab We will explore the remaining properties in the exercises. Section 2. 2 Matrices in Matlab 95 2. Exercises 1. Given the matrices A= and C= 3 1 , 5 8 3 3 , 2 1 B= 1 1 , 2 3 and C= 1 2 , 0 9 3. Given the matrices A= 1 0 , 2 5 B= 0 1 , 2 7 use Matlab to veri fy each of the following properties. Note that 0 represents the zero matrix. a) A + B = B + A b) (A + B) + C = A + (B + C) c) A + 0 = A d) A + (? A) = 0 use Matlab to verify each of the following forms of the distributive property. a) A(B + C) = AB + AC b) (A + B)C = AC + BC 4. Given the matrices A= 2 2 , 4 7 B= 3 1 , 8 9 2. The fact that matrix multiplication is not commutative is a huge loss. For example, with real numbers, the following familiar algeraic properties hold. . = ii. (a + b)2 = a2 + 2ab + b2 iii. (a + b)(a ? b) = a2 ? b2 Use Matlab and the matrices A= 1 1 4 2 and B = 2 3 1 6 (ab)2 a2 b2 and the scalars ? = 2 and ? = ? 3, use Matlab to verify each of the following properties. a) (? + ? )A = ? A + ? A b) ? (A + B) = ? A + ? B c) ? (? A) = ( )A d) (? A)B = ? (AB) = A(? B) 5. Enter the matrices A=pascal(3) and B=magic(3). a) Use Matlab to compute (A+B)T . b) Use Matlab to compute AT + B T and compare your result with the result from part (a). Explain what your learned in this exercise. to show that none of these properties is valid for these choices of A and B.Can you explain why each of properties (i-iii) is not valid for matrix multiplication? Hint: Try to expand the left-hand side of each property to arrive at the right-hand side. 96 Chapter 2 Vectors and Matrices in Matlab a) What is the result of the Matlab command A(:,2)=[ ]? Note: [ ] is the empty matrix. b) Refresh matrix A with A=pascal(4). What is the result of the Matlab command A(3,:)=[ ]? 13. Enter the matrix A=pascal(5). a) What command will add a row of all ones to the bottom of matrix A? Use Matlab to verify your conjecture. b) What command will add a column of all ones to the right end of matrix A?Use Matlab to verify your conjecture. 14. Enter the matrix A=magic(3). Execute the command A(5,4)=0. Explain the resulting matrix. 15. Enter the matrix A=ones(5). a) Explain how you can insert a row of all 5’s betwen rows 2 and 3 of matrix A. Use Matlab to verify your conjecure. b) Explain how you can insert a column of all 5’s betwen columns 3 and 4 of matrix A. Use Matlab to verify your conjecure. 16. Enter the matrix ? ? 1 2 3 A = ? 4 5 6?. 7 8 9 a) What is the output of the Matlab command A=A([1,3,2],:)? 6. Enter the matrix A=pascal(4) and the scalar ? = 5. a) Use Matlab to compute (? A)T . b) Use Matlab to compute ?A and compare your result with the result from part (a). Explain what your learned in this exercise. 7. Using hand calculations only, calculate the following matrix-vector product, then verify your result in Matlab. ? ? 1 1 2 1 ? 3 4 0 2 ? 0 5 6 ? 5 8. Write the following system of linear equations in matrix-vector form. 2x + 2y + 3z = ? 3 4x + 2y ? 8z = 12 3x + 2y + 5z = 10 9. Using hand calculations only, calculate the following matrix-matrix product, then verify your result in Matlab. ? ? 2 3 1 1 1 4 ? 0 1 2 0 0 5? 0 0 5 3 5 2 10. Enter the matrix magic(8). What Matlab command will zero out all of the even rows?Use Matlab to verify your conjecture. 11. Enter the matrix pascal(8). What Matlab command will zero out all of the odd columns? Use Matlab to verify your conjecture. 12. Enter the matrix A=pascal(4). Section 2. 2 b) Refresh matrix A to its original value. What Matlab command will swap columns 1 and 3 of matrix A? Use Matlab to verify your conjecture. 17. Enter the matrix ? ? 1 2 3 A = ? 4 5 6?. 7 8 9 a) Enter the Matlab command A(2,:)=A(2,:)-4*A(1,:)? Explain the result of this command. b) Continue with the resulting matrix A from part (a). What is the output of the Matlab command A(3,:)=A(3,:)-7*A(1,:)?Explain the result of this command. 18. Type format rat to change the display to rational format. Create a 3 ? 3 Hilbert matrix with the command H=hilb(3). a) What is the output of the Matlab command H(1,:)=6*H(:,1)? Explain the result of this command. b) Continue with the resulting matrix H from part (a). What command will clear the fractions from row 2 of this result? 19. Enter the matrices A=magic(3) and B=pascal(3). Execute the command C=A+i*B. Note: You may have to enter clear i to return i to its default (the square root of ? 1). a) What is the transpose of the matrix C? Use Matlab to verify your Matrices in Matlab 97 esponse. b) What is the conjugate transpose of the matrix C? Use Matlab to verify your response. 20. Use Matlab’s hadamard(n) command to form Hadarmard matrices of order n = 2, 4, 8, and 16. In each case, use Matlab to calculate H T H. Note the pattern. Explain in your own words what would happen if your formed the matrix product H T H, where H is a Hadamard matrix of order 256. 21. Enter the Matlab command magic(n) to form a â€Å"magic† matrix of order n = 8. Use Matlab’s sum command to sum both the columns and the rows of your â€Å"magic† matrix. Type help sum to learn how to use the syntax sum(X,dim) to accomplish this goal.What is â€Å"magic† about this matrix? 22. Enter the Matlab command A=magic(n) to form a â€Å"mag ic† matrix of order n = 8. Use Matlab’s sum command to sum the columns of your â€Å"magic† matrix. Explain how you can use matrix-vector multilication to sum the columns of matrix A. 23. Set A=pascal(5) and then set I=eye(5), then ? nd the matrix product AI. Why is I called the identity matrix? Describe what a 256 ? 256 identity matrix would look like. 24. Set A=pascal(4) and then set B=magic(4). What operation will produce the second column of the matrix product AB? Can this be done 98 Chapter 2 Vectors and Matrices in Matlab 28.Enter the Matlab command hankel(x) to form a Hankel matrix H, where x is the vector [1, 2, 3, 4]. The help ? le for the hankel commands describes the Hankel matrix as a symmetric matrix. Take the transpose of H. Describe what is mean by a symmetric matrix. 29. A Hilbert matrix H is de? ned by H(i, j) = 1/(i + j ? 1), where i ranges from 1 to the number of rows and j ranges from 1 to the number of columns. Use this de? nition and hand ca lculations to ? nd a Hilbert matrix of dimension 4 ? 4. Use format rat and Matlab’s hilb command to check your result. 30. The number of ways to choose k objects from a set of n objects is de? ed and calcualted with the formula n k = n! . k! (n ? k)! without ? nding the product AB? 25. Set the vector v=(1:5). ’ and the vector w=(2:6). ’. a) The product vT w is called an inner product because of the position of the transpose operator. Use Matalb to compute the inner product of the vectors v and w. b) The product vwT is called an outer product because of the position of the transpose operator. Use Matalb to compute the outer product of the vectors v and w. 26. Enter A=[0. 2 0. 6;0. 8 0. 4]. Calculate An for n = 2, 3, 4, 5, etc. Does this sequence of matrices converge? If so, to what approximate matrix do they converge? 7. Use Matlab ones command to create the matrices ? ? 2 2 2 1 1 A= , B = ? 2 2 2? , 1 1 2 2 2 and 3 3 . 3 3 Craft a Matlab command that will build the block diagonal matrix ? ? A 0 0 C = ? 0 B 0 ? , 0 0 C where the zeros in this matrix represent matrices of zeros of the appropriate size. De? ne a Pascal matrix P with the formula P (i, j) = i+j? 2 , i? 1 where i ranges from 1 to the number of rows and j ranges from 1 to the number of columns. Use this de? nition and hand calculations to ? nd a Pascal matrix of dimension 4 ? 4. Use Matlab’s pascal command to check your result. Section 2. 2 Matrices in Matlab 99 2. 2 Answers 1. ) Enter the matrices. gt;gt; A=[3 3;2 1]; B=[1 1;2 3]; Calculate A + B. gt;gt; A+B ans = 4 4 gt;gt; A+(B+C) ans = 7 5 9 12 c) Enter the matrix A and the zero matrix. gt;gt; A=[3 3;2 1]; O=zeros(2,2); 4 4 Calculate A + 0. gt;gt; A+O ans = 3 2 Calculate A. gt;gt; A A = 3 2 3 1 Calculate B + A. gt;gt; B+A ans = 4 4 3 1 4 4 b) Enter the matrices gt;gt; A=[3 3;2 1]; B=[1 1;2 3]; gt;gt; C=[3 1;5 8]; Calculate (A + B)C. gt;gt; (A+B)+C ans = 7 5 9 12 Calculate AC + BC. d) Enter the matrix A and the zero mat rix. startMatlab gt;gt; A=[3 3;2 1]; O=zeros(2,2); Calculate A + (? A). 100 Chapter 2 Vectors and Matrices in Matlab gt;gt; A+(-A) ans = 0 0 0 0 Calculate the zero matrix. gt;gt; O O = 0 0 0 0 5. gt;gt; (A+B)*C ans = 1 11 4 116 gt;gt; A*C+B*C ans = 1 11 4 116 a) Ener the matrices A and B. gt;gt; A=pascal(3); B=magic(3); Compute (A + B)T . 3. a) Enter the matrices A, B, and C. gt;gt; A=[1 0;2 5]; B=[0 1;2 7]; gt;gt; C=[1 2;0 9]; Compare A(B + C) and AB + AC. gt;gt; A*(B+C) ans = 1 3 12 86 gt;gt; A*B+A*C ans = 1 3 12 86 b) Enter the matrices A, B, and C. gt;gt; A=[1 0;2 5]; B=[0 1;2 7]; gt;gt; C=[1 2;0 9]; Compare (A+B)C and AC +BC. gt;gt; (A+B). ’ ans = 9 4 2 7 7 10 b) Compute AT + B T . gt;gt; A. ’+B. ’ ans = 9 4 2 7 7 10 12 8 5 12 8 The transpose of the sum of two matrices is equal to the sum of the transposes of the two matrices. 7. Enter matrix A and vector x. Section 2. 2 13. gt;gt; A=[1 1 2;3 4 0;0 5 6]; gt;gt; x=[1 2 5]. ’; Calculate Ax. gt;gt; A*x ans = 13 11 40 Matrices in Matlab 101 a) Enter the matrix A. gt;gt; A=pascal(5) To add a row of all ones to the bottom of the matrix, execute the following command. gt;gt; A(6,:)=ones(5,1) b) Enter the matrix A. gt;gt; A=pascal(5) To add a column of all ones to the right end of the matrix, execute the following command. gt;gt; A(:,6)=ones(5,1) 9. Enter matrices A and B. gt;gt; A=[2 3 1;0 1 2;0 0 5]; gt;gt; B=[1 1 4;0 0 5;3 5 2]; Calculate AB. gt;gt; A*B ans = 5 6 15 7 10 25 25 9 10 15. a) Enter the matrix A. 11. Enter matrix A. gt;gt; A=pascal(8); The following command will zero out all the odd columns. gt;gt; A(:,1:2:end)=0; gt;gt; A=ones(5); We’ll build a new matrix using the ? rst two roes of matrix A, then a row of 5’s, then the last three rows of matrix A. Note that we separate new columns with commas. 102 Chapter 2 Vectors and Matrices in Matlab gt;gt; B=[A(1:2,:);5*ones(1,5); A(3:5,:)] B = 1 1 1 1 1 1 1 1 1 1 5 5 5 5 5 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 b) Enter the matrix A. gt;gt; A=ones(5); We’ll build a new matrix using the ? rst 3 columns of matrix A, then a column of 5’s, then the last two columns of matrix A. Note that we separate new rows with semicolons. gt;gt; B=[A(:,1:3),5*ones(5,1), A(:,4:5)] B = 1 1 1 5 1 1 1 1 1 5 1 1 1 1 1 5 1 1 1 1 1 5 1 1 1 1 1 5 1 1 gt;gt; A=[1 2 3;4 5 6;7 8 9] A = 1 2 3 4 5 6 7 8 9 The next command will subtract 4 times row 1 from row 2. gt;gt; A(2,:)=A(2,:)-4*A(1,:) A = 1 2 3 0 -3 -6 7 8 9 b) Continuing with the last value of matrix A, the next command will subtract 7 times row 1 from row 3. gt;gt; A(3,:)=A(3,:)-7*A(1,:) A = 1 2 3 0 -3 -6 0 -6 -12 9. a) Enter the matrices A and B and compute C. gt;gt; A=magic(3); B=pascal(3); gt;gt; C=A+i*B The transpose of ? ? 8+i 1+i 6+i C = ? 3 + 1 5 + 2i 7 + 3i ? 4 + i 9 + 3i 2 + 6i 17. a) Enter the matrix A. Section 2. 2 is 8+i 3+i 4+i ? 1 + i 5 + 2i 9 + 3i ? . C = 6 + i 7 + 3i 2 + 6i T Matrices in Matlab 103 ? ? second dimension with the following command . You’ll note that the sum of each row is 260. gt;gt; sum(A,2) ans = 260 260 260 260 260 260 260 260 This result is veri? ed with the following Matlab command. gt;gt; C. ’ b) The conjugate ? 8+i ? 3 + 1 C= 4+i is 8? i 3? i 4? i ? 1 ? i 5 ? 2i 9 ? 3i ? . C = 6 ? 7 ? 3i 2 ? 6i T transpose of ? 1+i 6+i 5 + 2i 7 + 3i ? 9 + 3i 2 + 6i ? ? 23. Store A with the following command. gt;gt; A=pascal(5) A = 1 1 1 1 2 3 1 3 6 1 4 10 1 5 15 This result is veri? ed with the following Matlab command. gt;gt; C’ 1 4 10 20 35 1 5 15 35 70 21. Enter matrix A. gt;gt; A=magic(8) You sum the rows along the ? rst dimension with the following command. You’ll note that the sum of each column is 260. gt;gt; sum(A,1) You sum the columns along the You store I with the following command. gt;gt; I=eye(5) I = 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 Note that AI is identical to matrix A. 04 Chapter 2 Vectors and Matrices in Matlab You should be able to compute vwT manually and g et the same result. gt;gt; A*I ans = 1 1 1 1 1 1 2 3 4 5 1 3 6 10 15 1 4 10 20 35 1 5 15 35 70 27. Load the matrices A and B. gt;gt; A=ones(2); B=2*ones(3); Load the matrix C. gt;gt; C=3*ones(2);You can construct the required matrix with the following command. gt;gt; D=[A,zeros(2,3),zeros(2,2); zeros(3,2), B, zeros(3,2); zeros(2,2), zeros(2,3), C] A 256? 256 identity matrix would have 1’s on its main diagonal and zeros in all other entries. 25. a) Store the vectors v and w. gt;gt; v=(1:5). ’; w=(2:6). ; The inner product vT w is computed as follows. gt;gt; v. ’*w ans = 70 You should be able to compute vT w manually and get the same result. b) The outer product vwT is computed as follows. gt;gt; v*w. ’ ans = 2 3 4 6 6 9 8 12 10 15 29. The entry in row 1 column 1 would be H(1, 1) = 1/(1 + 1 ? 1) = 1. The entry in row 1 column 2 would be H(1, 2) = 1/(1+2? 1) = 1/2. Continuing in this manner, we arrive at a 4 ? 4 Hilbert matrix. ? ? 1 1/2 1/3 1/4 ? 1/2 1/3 1/ 4 1/5 ? H=? ? 1/3 1/4 1/5 1/6 1/4 1/5 1/6 1/7 This result can be veri? ed by these commands. 4 8 12 16 20 5 10 15 20 25 6 12 18 24 30 gt;gt; format rat gt;gt; H=hilb(4)