matrix-operations.wiki (3454B)
1 == Matrix operations == 2 $a_{ij}$ is the entry in the ith row and jth column of A 3 4 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. 5 6 equal matrices have same size _and_ their corresponding entries are equal. 7 8 === Sums and scalar multiples === 9 sum A+B: sum corresponding entries in A and B. 10 11 scalar multiple $rA$ is matrix whose columns are r times the corresponding columns in A (with r scalar). 12 13 the usual rules of algebra apply to sums and scalar multiples of matrices. 14 15 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. 16 17 $A(Bx) = (AB)x$ 18 19 $AB = A \begin{bmatrix} b_1 & b_2 & \dots & b_p \end{bmatrix} = \begin{bmatrix} Ab_1 & Ab_2 & \dots & Ab_p \end{bmatrix}$ 20 21 A is matrix, B is matrix with columns $b_1 \dots b_p$. 22 23 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. 24 25 if product AB is defined, then: 26 27 28 $(AB)_{ij} = a_{i1} b_{1j} + a_{i2} b_{2j} + \dots + a_{in} b_{nj}$ 29 30 $row_i (AB) = row_i (A) \times B$ 31 32 a nifty trick: if multiplying matrix m×n by matrix n×o, the product will be a matrix m×o. 33 34 === Powers of a matrix === 35 $A^k = \underbrace{A \dots A}_{k}$ 36 37 with $A$ an n × n matrix and k a positive integer. 38 39 === Transpose of a matrix === 40 a matrix $A'$ whose columns are made up of the corresponding rows of $A$ 41 42 properties: 43 * $(A^T)^T = A$ 44 * $(A+B)^T = A^T + B^T$ 45 * $(rA)^T = rA^T$ with r a scalar 46 * $(AB)^T = B^T A^T$$ 47 48 the transpose of a product of matrices == product of their transposes in reverse order 49 50 === Inverse of a matrix === 51 invertible (singular) if there is same size matrix C such that $CA = I$ and $AC = I$ where I is the n × n identity matrix. 52 53 identity matrix: a matrix where the diagonals are all 1. 54 55 C is uniquely determined by A, so: $A^{-1} A = I$. 56 57 let $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}.$ if $ad - bc \ne 0$ then $A^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$ 58 59 determinant: $\det A = ad - bc$ 60 61 if A is invertible (determinant is not 0), then for each $b \in \Re^n$ the solution of $Ax = b$ is $A^{-1} b$. 62 63 properties of inverse: 64 * $(A^{-1})^{-1} = A$ 65 * $(AB)^{-1} = B^{-1} A^{-1}$ (watch out for order!) 66 * $(A^T)^{-1} = (A^{-1})^T$ 67 68 finding $A^{-1}$: 69 * Row reduce augmented matrix $\begin{bmatrix} A & I \end{bmatrix}$. 70 * if A is row equivalent to I, then $\begin{bmatrix} A & I \end{bmatrix}$ is row equivalent to $\begin{bmatrix} I & A^{-1} \end{bmatrix}$ 71 * otherwise, A doesn't have an inverse. 72 73 === Elementary matrices === 74 elementary matrix: obtained by performing single elementary row operation on identity matrix 75 76 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 $I_m$ 77 78 inverse of any elementary matrix E is of same type that transforms E back into I. 79 80 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 $A^{-1}$. 81