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6.2 - Operations with Matrices
Course: MAT 1033, Spring 2011School: Valencia
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version="1.0" <?xml encoding="utf-8"?> <!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> <html xmlns="http://www.w3.org/1999/xhtml"> <head> <title>6.2 - Operations with Matrices</title> <meta http-equiv="Content-Type" content="text/html; charset=utf-8" /> <link href="../m116.css" rel="stylesheet" type="text/css" /> <style type="text/css"> <!-.inline { vertical-align: middle; } table { padding-top: 0.5em; padding-bottom: 0.5em; margin-top: 0.5em; margin-bottom: 0.5em; } --> </style> <link href="matrix.css" rel="stylesheet" type="text/css" /> </head> <body> <h1>6.2 - Operations with Matrices</h1> <h2>Equality</h2> <p>Two matrices are equal if and only if </p> <ul> <li>The order of the matrices are the same</li> <li>The corresponding elements of the matrices are the same</li> </ul> <h2>Addition</h2> <ul> <li>Order of the matrices must be the same</li> <li>Add corresponding elements together</li> <li>Matrix addition is commutative</li> <li>Matrix addition is associative</li> </ul> <h2>Subtraction</h2> <ul> <li>The order of the matrices must be the same</li> <li>Subtract corresponding elements</li> <li>Matrix subtraction is not commutative (neither is subtraction of real numbers)</li> <li>Matrix subtraction is not associative (neither is subtraction of real numbers)</li> </ul> <h2>Scalar Multiplication</h2> <p>A scalar is a number, not a matrix.</p> <ul><li>The matrix can be any order</li> <li>Multiply all elements in the matrix by the scalar</li> <li>Scalar multiplication is commutative</li> <li>Scalar multiplication is associative</li> </ul> <h2>Zero Matrix</h2>
<ul> <li>Matrix of any order</li> <li>Consists of all zeros</li> <li>Denoted by capital O</li> <li>Additive Identity for matrices</li> <li>Any matrix plus the zero matrix is the original matrix</li> </ul> <h2>Matrix Multiplication</h2> <p>A<sub>m×n</sub> × B<sub>n×p</sub> = C<sub>m×p</sub></p> <ul> <li>The number of columns in the first matrix must be equal to the number of rows in the second matrix. That is, the inner dimensions must be the same.</li> <li>The order of the product is the number of rows in the first matrix by the number of columns in the second matrix. That is, the dimensions of the product are the outer dimensions.</li> <li>Since the number of columns in the first matrix is equal to the number of rows in the second matrix, you can pair up entries.</li> <li>Each element in row <em>i</em> from the first matrix is paired up with an element in column <em>j</em> from the second matrix.</li> <li>The element in row <em>i</em>, column <em>j</em>, of the product is formed by multiplying these paired elements and summing them.</li> <li>Each element in the product is the sum of the products of the elements from row <em>i</em> of the first matrix and column <em>j</em> of the second matrix. </li> <li>There will be <em>n</em> products which are summed for each element in the product.</li> </ul> <p>See a complete example of <a href="multiplication.html">matrix multiplication</a>.</p> <h3>Matrix multiplication is not commutative</h3> <ul> <li>Multiplication of real numbers is.</li> <li>The inner dimensions may not agree if the order of the matrices is changed.</ li> </ul> <h3>Do not simply multiply corresponding elements together</h3> <ul> <li>Since the order (dimensions) of the matrices don't have to be the same, there may not be corresponding elements to multiply together.</li> <li>Multiply the rows of the first by the columns of the second and add.</li> </ul> <h3>There is no matrix division</h3> <ul> <li>There is no defined process for dividing a matrix by another matrix.</li> <li>A matrix may be divided by a scalar.</li> </ul> <h2>Identity Matrix</h2>
<ul> <li>Square matrix</li> <li>Ones on the main diagonal</li> <li>Zeros everywhere else</li> <li>Denoted by I. If a subscript is included, it is the order of the identity matrix.</li> <li>I is the multiplicative identity for matrices</li> <li>Any matrix times the identity matrix is the original matrix.</li> <li>Multiplication by the identity matrix is commutative, although the order of the identity may change</li> </ul> <p>Identity matrix of size 2 </p> <table border="0" cellspacing="0" cellpadding="1"> <tr> <td rowspan="2" class="downhalf">I<sub>2</sub> =</td> <td class="matlt"> </td> <td class="lhs">1</td> <td class="lhs">0</td> <td class="matrt"> </td> </tr> <tr> <td class="matlb"> </td> <td class="lhs">0</td> <td class="lhs">1</td> <td class="matrb"> </td> </tr> </table> <p>Identity matrix of size 3</p> <table border="0" cellspacing="0" cellpadding="1"> <tr> <td rowspan="3" class="downhalf">I<sub>3</sub> = </td> <td class="matlt"> </td> <td class="lhs">1</td> <td class="lhs">0</td> <td class="lhs">0</td> <td class="matrt"> </td> </tr> <tr> <td class="matlm"> </td> <td class="lhs">0</td> <td class="lhs">1</td> <td class="lhs">0</td> <td class="matrm"> </td> </tr> <tr> <td class="matlb"> </td> <td class="lhs">0</td> <td class="lhs">0</td> <td class="lhs">1</td> <td class="matrb"> </td> </tr> </table> <h2>Properties of Matrices</h2> <table cellpadding="3" cellspacing="0"> <tr> <th scope="col" class="datath">Property</th>
<th scope="col" class="datath">Example</th> </tr> <tr> <td class="datal">Commutativity of Addition</td> <td class="datal">A + B = B + A</td> </tr> <tr> <td class="datal">Associativity of Addition</td> <td class="datal">A + ( B + C ) = ( A + B ) + C</td> </tr> <tr> <td class="datal">Associativity of Scalar Multiplication</td> <td class="datal">(cd) A = c (dA)</td> </tr> <tr> <td class="datal">Scalar Identity</td> <td class="datal">1A = A(1) = A</td> </tr> <tr> <td class="datal">Distributive</td> <td class="datal">c (A + B) = cA + cB</td> </tr> <tr> <td class="datal">Distributive</td> <td class="datal">(c + d) A = cA + dA</td> </tr> <tr> <td class="datal">Additive Identity</td> <td class="datal">A + O = O + A = A</td> </tr> <tr> <td class="datal">Associativity of Multiplication</td> <td class="datal">A (BC) = (AB) C</td> </tr> <tr> <td class="datal">Left Distributive</td> <td class="datal">A (B + C) = AB + AC</td> </tr> <tr> <td class="datal">Right Distributive</td> <td class="datal">( A + B ) C = AC + BC</td> </tr> <tr> <td class="datal">Scalar Associativity / Commutativity</td> <td class="datal">c (AB) = (cA) B = A (cB) = (AB) c</td> </tr> <tr> <td class="datal">Multiplicative Identity</td> <td class="datal">IA = AI = A</td> </tr></table> <h2>Properties of Real Numbers that aren't Properties of Matrices</h2> <h3>Commutativity of Multiplication</h3> <ul> <li>You can not change the order of a multiplication problem and expect to get the same product. AB≠BA</li> <li>You must be careful when factoring common factors to make sure they are on the
same side. AX+BX = (A+B)X XA and + XB = X(A+B) but AX + XB doesn't factor.</li> </ul> <h3>Zero Product Property</h3> <ul> <li>Just because a product of two matrices is the zero matrix does not mean that one of them was the zero matrix.</li> </ul> <h3>Multiplicative Property of Equality</h3> <ul> <li>If A=B, then AC = BC. This property is still true, but the converse is not necessarily true. Just because AC = BC does not mean that A = B.</li> <li>Because matrix multiplication is not commutative, you must be careful to either pre-multiply or post-multiply on both sides of the equation. That is, if A=B, then AC = BC or CA = CB, but AC≠CB.</li> </ul> <h3>There is no matrix division</h3> <ul> <li>You must multiply by the inverse of the matrix</li> </ul> <h2>Evaluating a Function using a Matrix</h2> <p>Consider the function f(x) = x<sup>2</sup> - 4x + 3 and the matrix A</p> <table border="0" cellspacing="0" cellpadding="1"> <tr> <td rowspan="2" class="downhalf">A =</td> <td class="matlt"> </td> <td class="lhs">1</td> <td class="lhs">2</td> <td class="matrt"> </td> </tr> <tr> <td class="matlb"> </td> <td class="lhs">3</td> <td class="lhs">4</td> <td class="matrb"> </td> </tr> </table> <p>The initial attempt to evaluate the f(A) would be to replace every x with an A to get f(A) = A<sup>2</sup> - 4A + 3. There is one slight problem, however. The constant 3 is not a matrix, and you can't add matrices and scalars together. So, we multiply the constant by the Identity matrix.</p> <p>f(A) = A<sup>2</sup> - 4A + 3I.</p> <p>Evaluate each term in the function and then add them together.</p> <table border="0" cellspacing="0" cellpadding="1"> <tr> <td rowspan="2" class="downhalf">A<sup>2</sup> =</td> <td class="matlt"> </td> <td class="lhs">1</td> <td class="lhs">2</td> <td class="matrt"> </td> <td rowspan="2" class="downhalf">*</td> <td class="matlt"> </td> <td class="lhs">1</td> <td class="lhs">2</td> <td class="matrt"> </td>
<td rowspan="2" class="downhalf">=</td> <td class="matlt"> </td> <td class="lhs">7</td> <td class="lhs">10</td> <td class="matrt"> </td> </tr> <tr> <td class="matlb"> </td> <td class="lhs">3</td> <td class="lhs">4</td> <td class="matrb"> </td> <td class="matlb"> </td> <td class="lhs">3</td> <td class="lhs">4</td> <td class="matrb"> </td> <td class="matlb"> </td> <td class="lhs">15</td> <td class="lhs">22</td> <td class="matrb"> </td> </tr> </table> <table border="0" cellspacing="0" cellpadding="1"> <tr> <td rowspan="2" class="downhalf">-4 A = -4</td> <td class="matlt"> </td> <td class="lhs">1</td> <td class="lhs">2</td> <td class="matrt"> </td> <td rowspan="2" class="downhalf">=</td> <td class="matlt"> </td> <td class="lhs">-4</td> <td class="lhs">-8</td> <td class="matrt"> </td> </tr> <tr> <td class="matlb"> </td> <td class="lhs">3</td> <td class="lhs">4</td> <td class="matrb"> </td> <td class="matlb"> </td> <td class="lhs">-12</td> <td class="lhs">-16</td> <td class="matrb"> </td> </tr> </table> <table border="0" cellspacing="0" cellpadding="1"> <tr> <td rowspan="2" class="downhalf">3I = 3</td> <td class="matlt"> </td> <td class="lhs">1</td> <td class="lhs">0</td> <td class="matrt"> </td> <td rowspan="2" class="downhalf">=</td> <td class="matlt"> </td> <td class="lhs">3</td> <td class="lhs">0</td> <td class="matrt"> </td>
</tr> <tr> <td class="matlb"> </td> <td class="lhs">0</td> <td class="lhs">1</td> <td class="matrb"> </td> <td class="matlb"> </td> <td class="lhs">0</td> <td class="lhs">3</td> <td class="matrb"> </td> </tr> </table> <table border="0" cellspacing="0" cellpadding="1"> <tr> <td rowspan="2" class="downhalf">f(A) =</td> <td class="matlt"> </td> <td class="lhs">7</td> <td class="lhs">10</td> <td class="matrt"> </td> <td rowspan="2" class="downhalf">+</td> <td class="matlt"> </td> <td class="lhs">-4</td> <td class="lhs">-8</td> <td class="matrt"> </td> <td rowspan="2" class="downhalf">+</td> <td class="matlt"> </td> <td class="lhs">3</td> <td class="lhs">0</td> <td class="matrt"> </td> <td rowspan="2" class="downhalf">=</td> <td class="matlt"> </td> <td class="lhs">6</td> <td class="lhs">2</td> <td class="matrt"> </td> </tr> <tr> <td class="matlb"> </td> <td class="lhs">15</td> <td class="lhs">22</td> <td class="matrb"> </td> <td class="matlb"> </td> <td class="lhs">-12</td> <td class="lhs">-16</td> <td class="matrb"> </td> <td class="matlb"> </td> <td class="lhs">0</td> <td class="lhs">3</td> <td class="matrb"> </td> <td class="matlb"> </td> <td class="lhs">3</td> <td class="lhs">9</td> <td class="matrb"> </td> </tr> </table> <h2>Factoring Expressions</h2> <p>Some examples of factoring are shown. Simplify and solve like normal, but remember that matrix
multiplication is not commutative and there is no matrix division.</p> <p class="singlespace">2X + 3X = 5X </p> <p class="singlespace">AX + BX = (A+B)X </p> <p class="singlespace">XA + XB = X(A+B) </p> <p class="singlespace">AX + 5X = (A+5I)X </p> <p class="singlespace">AX+XB does not factor</p> <h2>Solving Equations</h2> <p>A system of linear equations can be written as AX=B where A is the coefficient matrix, X is a column vector containing the variables, and B is the right hand side. We'll learn <a href="inverses.html">how to solve this equation</a> in the next section.</p> <p>If there are more than one system of linear equations with the same coefficient matrix, then you can expand the B matrix to have more than one column. Put each right hand side into its own column.</p> <h2>Matrix Multiplication</h2> <p>Matrix multiplication involves summing a product. It is appropriate where you need to multiply things together and then add. As an example, multiplying the number of units by the per unit cost will give the total cost.</p> <p>The units on the product are found by performing unit analysis on the matrices. The labels for the product are the labels of the rows of the first matrix and the labels of the columns of the second matrix.</p> </body> </html>
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Is the expected return on a stock with a beta=2.0 twice the expected return on a stock with abeta=1.0?Perhaps the single most important measure of stock risk or volatility is a stock's beta. It'sone of those at-a-glance measures that can provide seriou
Berklee - FIN - 550
A firm issues a bond at par value. Shortly thereafter, interest rates fall. If you calculated thecoupon rate, coupon yield, and yield to maturity for this bond after the decline in interestrates, which of the three values would be highest and which woul
Berklee - FIN - 550
When using discounted cash flow analysis to value an asset, explain why it is important tomeasure the risk of the asset and to associate an expected return with that risk measure.The procedure for valuing a risky asset involves three basic steps: (1) de
Berklee - FIN - 550
Suppose there are two risky assets. One offers a higher return than the other, but it also has ahigher standard deviation. Will one of these assets always lie on the efficient frontier? Will oneof them always be inefficient if held alone?Risky asset is
Berklee - FIN - 550
Suppose that a mutual fund has a beta equal to 0.75. Is it necessarily the case that the standarddeviation of returns on the fund is less than the standard deviation of market returns?A mutual fund is a type of an investment company that pools money fro
Berklee - FIN - 550
Why is using the cost of equity to discount project cash flows inappropriate when a firm usesboth debt and equity in its capital structure?Capital structure is the manner in which a firms assets are financed; that is, the right-handside of the balance
Berklee - FIN - 550
Why would a project that reaches the break-even point in terms of net income potentially bebad for shareholders?The EBIT-EPS capital structure approach focuses on finding a capital structure with thehighest EPS (earnings per share) over the expected ra
Berklee - FIN - 550
Describe how the IRR and NPV approaches are related.The internal rate of return and the net present value methods are the two major approachesto evaluating capital budgeting projects. The NPV technique measures the present value of thefuture cash flows
Berklee - FIN - 550
Why do we consider changes in net working capital associated with a project to be cashinflows or outflows?Working capital is useful to show the operating liquidity of a company and how thecompany manages its business. Positive working capital is requir
Berklee - FIN - 550
Two firms in the same industry have very different equity betas. Offer two reasons why thiscan occur?Beta is an indirect measure, which compares the systematic risk associated with acompanys shares with the systematic risk of the capital market as a wh
Berklee - FIN - 550
Why must manager intuition be part of the investment decision process regardless of aproject's NPV or IRR?Making decisions is certainly the most important task of a manager and it is often a verydifficult one. Experienced managers rarely make major inv
Berklee - FIN - 550
What does it mean if a project has an NPV of $1 million? Explain.In finance, the net present value (NPV) or net present worth (NPW) of a time series of cashflows, both incoming and outgoing, is defined as the sum of the present values (PVs) of theindiv
Berklee - FIN - 550
For what kinds of investments would terminal value account for a substantial fraction of thetotal project NPV, and for what kinds of investments would terminal value be relativelyunimportant?Some investments have a well-defined life span. The life span
Berklee - FIN - 550
Suppose an analyst makes a mistake and calculates the NPV or an investment project bydiscounting the project's contribution to net income each year rather than by discounting itscash flow. Would you expect the NPV based on net income to be higher or low
Berklee - FIN - 550
What is an event study designed to test?In disputes that involve publicly traded companies, a statistical technique known as an eventstudy, which can be used to relate security price movements to specific events, is an important partof the economic exp
Berklee - FIN - 550
What role does par value play in the pricing and sale of common stock by the issuingcorporation? Why do most firms assign relatively low par values to their shares?Par value, sometimes referred to as face value, is the nominal value assigned to anunder