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My math cheat sheet

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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.


\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}

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