My math cheat sheet

## My math cheat sheet

This is a small table of fundamental math truths that are difficult for me to remember, since they are not my daily work, yet I need to recall them from time to time.

### Matrixes

\begin{align} C= \begin{bmatrix} a \\ b \end{bmatrix} \space \space \space \space \space \space \space \space R= \begin{bmatrix} c && d \end{bmatrix} \end{align} \begin{align} CR = \begin{bmatrix} a \\ b \end{bmatrix} \begin{bmatrix} c && d \end{bmatrix} = \begin{bmatrix} ac && ad \\ bc && bd \end{bmatrix} \end{align} \begin{align} RC = \begin{bmatrix} c && d \end{bmatrix} \begin{bmatrix} a \\ b \end{bmatrix} = \begin{bmatrix} ac+bd \end{bmatrix} \end{align}

$CC$ and $RR$ do not exist.
$CC^T$ and $R^TR$ result in 1x1 matrixes.
$C^TC$ and $RR^T$ result in 2x2 matrixes.

CRRC rule: # columns in first matrix must match the # rows of second matrix. The result matrix has the # rows of first matrix and # columns of the second.

Another mnemonic: $a,b=c,d$ while $a,b$ and $c,d$ are the sizes (rows, cols) of 1st and 2nd matrix respectively. Multiplication is defined when $b=c$, and size of result is $a,d$. \begin{align} C= \begin{bmatrix} a \\ b \\ c \end{bmatrix} \space \space \space \space \space \space \space \space R= \begin{bmatrix} d && e \end{bmatrix} \end{align}

In this case, $CR$ is defined, but $RC$ is not because 2 columns in R do not match 3 lines in C.

### Discrete transforms

Laplace transform $L(s)$, Fourier transform $F(w)$ and Z-transform $X(z)$ are related, being the Z-transform the most general.

\begin{align} X(z) = \sum\limits_{n=0}^\infty x_n z^{-n} \end{align} \begin{align} L(s) = X^*(s) = X(e^s) = \sum\limits_{n=0}^\infty x_n e^{-sn} \end{align} \begin{align} F(w) = L(jw) = X(e^{jw}) = \sum\limits_{n=0}^\infty x_n e^{-jwn} \end{align}