6.4 Basic matrix arthimetic

Some basic rules, for suitably-sized matrices $A$ and $B$ :

$A + B = B + A$ .
$c A$ for some constant $c$ multiplies every element of matrix $A$ by $c$ .
$A B \ne B A$ in general.
$(A B)^T = B^T A^T$ .
$(A B)^{-1} = B^{-1} A^{-1}$ .
$I A = A I = A$ for a suitably-sized identity matrix $I$ .
If $A$ is symmetric, then $A^{-1}$ (if it exists) is also symmetric.

Two specific cases sometimes come in handy. Consider a vector $\textbf{y}$ of length $n$ (a matrix of size $n\times 1$ ): $\textbf{y} = \left[ \begin{array}{c} y_1\\ y_2\\ y_3\\ \vdots\\ y_n \end{array} \right].$ Then multiplying $\textbf{y}^T$ by $\textbf{y}$ produces a $1\times 1$ vector (a scalar): $\begin{aligned} \textbf{y}^T\textbf{y} &= \left[ \begin{array}{ccccc} y_1\quad& y_2\quad& y_3\quad& \cdots\quad& y_n \end{array} \right] \times \left[ \begin{array}{c} y_1\\ y_2\\ y_3\\ \vdots\\ y_n \end{array} \right]\\ &= y_1^2 + y_2^2 + \cdots + y_n^2 = \sum_{i=1}^n y^2_i. \end{aligned}$ Also, $\begin{aligned} \textbf{y}\textbf{y}^T &= \left[ \begin{array}{c} y_1\\ y_2\\ y_3\\ \vdots\\ y_n \end{array} \right] \times \left[ \begin{array}{ccccc} y_1\quad& y_2\quad& y_3\quad& \cdots\quad& y_n \end{array} \right]\\ &= \left[ \begin{array}{cccc} y_1^2 & y_1 y_2 & \dots & y_1 y_n \\ y_2 y_1 & y_2^2 & \dots & y_2 y_n\\ \vdots & \vdots & \ddots & \vdots\\ y_n y_1 & y_n y_2 & \dots & y_n^2 \end{array} \right]. \end{aligned}$