IMG_1743.HEIC

# Matrices Let $A$ be an $m \times n$ matrix: $A = \begin{bmatrix} a_{11} & a_{12} &... & a_{1n} \\ a_{21} & a_{22} &... & a_{2n} \\... &... &... &... \\ a_{m1} & a_{m2} &... & a_{mn} \end{bmatrix}$ $a_{ij}$: element on row $i$, column $j$ **Main diagonal elements**: $a_{11}, a_{22}, a_{33},...$...

# Matrices Let $A$ be an $m \times n$ matrix: $A = \begin{bmatrix} a_{11} & a_{12} &... & a_{1n} \\ a_{21} & a_{22} &... & a_{2n} \\... &... &... &... \\ a_{m1} & a_{m2} &... & a_{mn} \end{bmatrix}$ $a_{ij}$: element on row $i$, column $j$ **Main diagonal elements**: $a_{11}, a_{22}, a_{33},...$ ## Types of Matrices * **Row matrix**: $1 \times n$ * **Column matrix**: $m \times 1$ * **Square matrix**: $m = n$ * **Zero matrix**: all elements are 0 * **Identity matrix**: * Square matrix * All main diagonal elements are 1 * All other elements are 0 $I_3 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$ * **Diagonal matrix**: * Square matrix * All elements not on the main diagonal are 0 $A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{bmatrix}$ * **Triangular matrix**: * **Upper triangular matrix**: All elements below the main diagonal are 0 $A = \begin{bmatrix} 1 & 2 & 3 \\ 0 & 4 & 5 \\ 0 & 0 & 6 \end{bmatrix}$ * **Lower triangular matrix**: All elements above the main diagonal are 0 $A = \begin{bmatrix} 1 & 0 & 0 \\ 2 & 3 & 0 \\ 4 & 5 & 6 \end{bmatrix}$ * **Transpose of a matrix**: Rows become columns $A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}$ $A^T = \begin{bmatrix} 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{bmatrix}$ * **Symmetric matrix**: $A^T = A$ * **Anti-symmetric matrix**: $A^T = -A$ ## Matrix Operations Let $A$ and $B$ be $m \times n$ matrices: $A = \begin{bmatrix} a_{11} & a_{12} &... & a_{1n} \\ a_{21} & a_{22} &... & a_{2n} \\... &... &... &... \\ a_{m1} & a_{m2} &... & a_{mn} \end{bmatrix}$ $B = \begin{bmatrix} b_{11} & b_{12} &... & b_{1n} \\ b_{21} & b_{22} &... & b_{2n} \\... &... &... &... \\ b_{m1} & b_{m2} &... & b_{mn} \end{bmatrix}$ ### Equality $A = B$ if and only if $a_{ij} = b_{ij}$ ### Addition $A + B = \begin{bmatrix} a_{11} + b_{11} & a_{12} + b_{12} &... & a_{1n} + b_{1n} \\ a_{21} + b_{21} & a_{22} + b_{22} &... & a_{2n} + b_{2n} \\... &... &... &... \\ a_{m1} + b_{m1} & a_{m2} + b_{m2} &... & a_{mn} + b_{mn} \end{bmatrix}$ ### Scalar Multiplication Let $k$ be a scalar: $kA = \begin{bmatrix} ka_{11} & ka_{12} &... & ka_{1n} \\ ka_{21} & ka_{22} &... & ka_{2n} \\... &... &... &... \\ ka_{m1} & ka_{m2} &... & ka_{mn} \end{bmatrix}$ ### Multiplication Let $A$ be an $m \times n$ matrix and $B$ be an $n \times p$ matrix: $A = \begin{bmatrix} a_{11} & a_{12} &... & a_{1n} \\ a_{21} & a_{22} &... & a_{2n} \\... &... &... &... \\ a_{m1} & a_{m2} &... & a_{mn} \end{bmatrix}$ $B = \begin{bmatrix} b_{11} & b_{12} &... & b_{1p} \\ b_{21} & b_{22} &... & b_{2p} \\... &... &... &... \\ b_{n1} & b_{n2} &... & b_{np} \end{bmatrix}$ $AB = C = \begin{bmatrix} c_{11} & c_{12} &... & c_{1p} \\ c_{21} & c_{22} &... & c_{2p} \\... &... &... &... \\ c_{m1} & c_{m2} &... & c_{mp} \end{bmatrix}$ $c_{ij} = a_{i1}b_{1j} + a_{i2}b_{2j} +... + a_{in}b_{nj}$