matrix-operations.html (4757B)
1 <!DOCTYPE html> 2 <html> 3 <head> 4 <script type="text/javascript" async src="https://cdn.jsdelivr.net/gh/mathjax/MathJax@2.7.5/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script> 5 <link rel="Stylesheet" type="text/css" href="style.css"> 6 <title>matrix-operations</title> 7 <meta http-equiv="Content-Type" content="text/html; charset=utf-8"> 8 </head> 9 <body> 10 11 <div id="Matrix operations"><h2 id="Matrix operations">Matrix operations</h2></div> 12 <p> 13 \(a_{ij}\) is the entry in the ith row and jth column of A 14 </p> 15 16 <p> 17 diagonal entries are \(a_{11}\), \(a_{22}\), etc. and form the main diagonal. if non-diagonal entries are zero, then it's a diagonal matrix. 18 </p> 19 20 <p> 21 equal matrices have same size <em>and</em> their corresponding entries are equal. 22 </p> 23 24 <div id="Matrix operations-Sums and scalar multiples"><h3 id="Sums and scalar multiples">Sums and scalar multiples</h3></div> 25 <p> 26 sum A+B: sum corresponding entries in A and B. 27 </p> 28 29 <p> 30 scalar multiple \(rA\) is matrix whose columns are r times the corresponding columns in A (with r scalar). 31 </p> 32 33 <p> 34 the usual rules of algebra apply to sums and scalar multiples of matrices. 35 </p> 36 37 <p> 38 when matrix B multiplies vector x, it transforms x into vector \(Bx\). if \(Bx\) is multiplied by A, the result is \(A(Bx)\). \(A(Bx)\) is produced from x by composition of mappings, which can be represented as multiplication by a single matrix AB. 39 </p> 40 41 <p> 42 \(A(Bx) = (AB)x\) 43 </p> 44 45 <p> 46 \(AB = A \begin{bmatrix} b_1 & b_2 & \dots & b_p \end{bmatrix} = \begin{bmatrix} Ab_1 & Ab_2 & \dots & Ab_p \end{bmatrix}\) 47 </p> 48 49 <p> 50 A is matrix, B is matrix with columns \(b_1 \dots b_p\). 51 </p> 52 53 <p> 54 each column of AB is a linear combination of columns of A using weights from corresponding column of B. AB has the same number of rows as A and same number of columns as B. if the number of columns of A does not match number of rows of B, the product is undefined. in general, AB ≠ BA. 55 </p> 56 57 <p> 58 if product AB is defined, then: 59 </p> 60 61 62 <p> 63 \((AB)_{ij} = a_{i1} b_{1j} + a_{i2} b_{2j} + \dots + a_{in} b_{nj}\) 64 </p> 65 66 <p> 67 \(row_i (AB) = row_i (A) \times B\) 68 </p> 69 70 <p> 71 a nifty trick: if multiplying matrix m×n by matrix n×o, the product will be a matrix m×o. 72 </p> 73 74 <div id="Matrix operations-Powers of a matrix"><h3 id="Powers of a matrix">Powers of a matrix</h3></div> 75 <p> 76 \(A^k = \underbrace{A \dots A}_{k}\) 77 </p> 78 79 <p> 80 with \(A\) an n × n matrix and k a positive integer. 81 </p> 82 83 <div id="Matrix operations-Transpose of a matrix"><h3 id="Transpose of a matrix">Transpose of a matrix</h3></div> 84 <p> 85 a matrix \(A'\) whose columns are made up of the corresponding rows of \(A\) 86 </p> 87 88 <p> 89 properties: 90 </p> 91 <ul> 92 <li> 93 \((A^T)^T = A\) 94 95 <li> 96 \((A+B)^T = A^T + B^T\) 97 98 <li> 99 \((rA)^T = rA^T\) with r a scalar 100 101 <li> 102 \((AB)^T = B^T A^T\)$ 103 104 </ul> 105 106 <p> 107 the transpose of a product of matrices == product of their transposes in reverse order 108 </p> 109 110 <div id="Matrix operations-Inverse of a matrix"><h3 id="Inverse of a matrix">Inverse of a matrix</h3></div> 111 <p> 112 invertible (singular) if there is same size matrix C such that \(CA = I\) and \(AC = I\) where I is the n × n identity matrix. 113 </p> 114 115 <p> 116 identity matrix: a matrix where the diagonals are all 1. 117 </p> 118 119 <p> 120 C is uniquely determined by A, so: \(A^{-1} A = I\). 121 </p> 122 123 <p> 124 let \(A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}.\) if \(ad - bc \ne 0\) then \(A^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}\) 125 </p> 126 127 <p> 128 determinant: \(\det A = ad - bc\) 129 </p> 130 131 <p> 132 if A is invertible (determinant is not 0), then for each \(b \in \Re^n\) the solution of \(Ax = b\) is \(A^{-1} b\). 133 </p> 134 135 <p> 136 properties of inverse: 137 </p> 138 <ul> 139 <li> 140 \((A^{-1})^{-1} = A\) 141 142 <li> 143 \((AB)^{-1} = B^{-1} A^{-1}\) (watch out for order!) 144 145 <li> 146 \((A^T)^{-1} = (A^{-1})^T\) 147 148 </ul> 149 150 <p> 151 finding \(A^{-1}\): 152 </p> 153 <ul> 154 <li> 155 Row reduce augmented matrix \(\begin{bmatrix} A & I \end{bmatrix}\). 156 157 <li> 158 if A is row equivalent to I, then \(\begin{bmatrix} A & I \end{bmatrix}\) is row equivalent to \(\begin{bmatrix} I & A^{-1} \end{bmatrix}\) 159 160 <li> 161 otherwise, A doesn't have an inverse. 162 163 </ul> 164 165 <div id="Matrix operations-Elementary matrices"><h3 id="Elementary matrices">Elementary matrices</h3></div> 166 <p> 167 elementary matrix: obtained by performing single elementary row operation on identity matrix 168 </p> 169 170 <p> 171 if elementary row operation is performed on m × n matrix A, result is EA, with E an m × m matrix obtained by performing same row operation on \(I_m\) 172 </p> 173 174 <p> 175 inverse of any elementary matrix E is of same type that transforms E back into I. 176 </p> 177 178 <p> 179 an n² matrix A is only invertible if A is row equivalent to \(I_n\). any sequence of elementary operations reducing A to \(I_n\) also transforms \(I_n\) into \(A^{-1}\). 180 </p> 181 182 </body> 183 </html>