Matrices and arrays

Parentheses around a grid is the default matrix.

\begin{pmatrix}1 & 2 \\ 3 & 4\end{pmatrix}

(3.17)

Brackets when I mean a linear map written in coordinates:

\begin{bmatrix}𝑎 & 𝑏 \\ 𝑐 & 𝑑\end{bmatrix}

(3.18)

A brace is the same object with a different fence. I use it when I am about to write a piecewise linear map.

\begin{Bmatrix}1 & 2 \\ 3 & 4\end{Bmatrix}

(3.19)

The identity is the first $3\times 3$ I actually type:

\begin{pmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{pmatrix}

(3.20)

Entries that are not integers. If fractions in a cell collapse, the matrix is a lie.

\begin{pmatrix}1 & \frac{1}{2} & \frac{1}{3} \\ \frac{1}{2} & \frac{1}{3} & \frac{1}{4} \\ \frac{1}{3} & \frac{1}{4} & \frac{1}{5}\end{pmatrix}

(3.21)

A rotation in the plane, sines sitting next to cosines:

\begin{pmatrix}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix}

(3.22)

A column is a matrix with one column. I write vectors that way when I am about to multiply.

\begin{pmatrix}𝑥 \\ 𝑦 \\ 𝑧\end{pmatrix}

(3.23)

A row is the other way around:

\begin{pmatrix}𝑥 & 𝑦 & 𝑧\end{pmatrix}

(3.24)

Determinant bars are a delimiter, not a new object:

\begin{vmatrix}𝑎 & 𝑏 \\ 𝑐 & 𝑑\end{vmatrix}=𝑎𝑑-𝑏𝑐

(3.25)

The $3\times 3$ determinant I will actually expand, not a $2\times 2$ again:

\begin{vmatrix}𝑎 & 𝑏 & 𝑐 \\ 𝑑 & 𝑒 & 𝑓 \\ 𝑔 & h & 𝑖\end{vmatrix}=𝑎(𝑒𝑖-𝑓h)-𝑏(𝑑𝑖-𝑓𝑔)+𝑐(𝑑h-𝑒𝑔)

(3.26)

Binomial coefficients are the other array that is not a matrix. I need them for Taylor remainders.

\binom{𝑛}{𝑘}=\frac{𝑛!}{𝑘!(𝑛-𝑘)!}

(3.27)

The binomial theorem, because a sum of binomials is the reason I wanted the symbol:

{(𝑥+𝑦)}^{𝑛}=\sum _{𝑘=0}^{𝑛}\binom{𝑛}{𝑘}{𝑥}^{𝑛-𝑘}{𝑦}^{𝑘}

(3.28)

A $2\times 2$ block acting on a vector is the linear system from (3.5) written as one product:

\begin{pmatrix}{𝑥}^{\prime } \\ {𝑦}^{\prime }\end{pmatrix}=\begin{pmatrix}𝑎 & 𝑏 \\ 𝑐 & 𝑑\end{pmatrix}\begin{pmatrix}𝑥 \\ 𝑦\end{pmatrix}

(3.29)

The three-dimensional version, so a converter cannot special-case a $2\times 2$ :

\begin{pmatrix}{𝑥}^{\prime } \\ {𝑦}^{\prime } \\ {𝑧}^{\prime }\end{pmatrix}=\begin{pmatrix}{𝑎}_{11} & {𝑎}_{12} & {𝑎}_{13} \\ {𝑎}_{21} & {𝑎}_{22} & {𝑎}_{23} \\ {𝑎}_{31} & {𝑎}_{32} & {𝑎}_{33}\end{pmatrix}\begin{pmatrix}𝑥 \\ 𝑦 \\ 𝑧\end{pmatrix}

(3.30)

An augmented matrix, because a linear system is a matrix with a bar I did not invent:

\begin{bmatrix}1 & 2 & 3 & \text{|} & 1 \\ 0 & 1 & 4 & \text{|} & 0 \\ 0 & 0 & 1 & \text{|} & 2\end{bmatrix}

(3.31)

The inverse of a $2\times 2$ , written as a scalar times a matrix, not as a wish:

{\begin{pmatrix}𝑎 & 𝑏 \\ 𝑐 & 𝑑\end{pmatrix}}^{-1}=\frac{1}{𝑎𝑑-𝑏𝑐}\begin{pmatrix}𝑑 & -𝑏 \\ -𝑐 & 𝑎\end{pmatrix}

(3.32)

Transpose and a conjugate transpose on the same line:

𝐴=\begin{pmatrix}𝑎 & 𝑏 \\ 𝑐 & 𝑑\end{pmatrix},\quad {𝐴}^{𝑇}=\begin{pmatrix}𝑎 & 𝑐 \\ 𝑏 & 𝑑\end{pmatrix},\quad {𝐴}^{\ast }={({𝐴}^{𝑇})}^{\ast }

(3.33)

Jacobian of a map ${\mathbb{R}}^{2}\to {\mathbb{R}}^{2}$ . Mixed partials live in the cells.

𝐷𝑓=\begin{pmatrix}\frac{\partial {𝑓}_{1}}{\partial 𝑥} & \frac{\partial {𝑓}_{1}}{\partial 𝑦} \\ \frac{\partial {𝑓}_{2}}{\partial 𝑥} & \frac{\partial {𝑓}_{2}}{\partial 𝑦}\end{pmatrix}

(3.34)

Hessian of a scalar, so second derivatives have to sit in a grid:

𝐻𝑓=\begin{pmatrix}\frac{{\partial }^{2}𝑓}{\partial {𝑥}^{2}} & \frac{{\partial }^{2}𝑓}{\partial 𝑥\partial 𝑦} \\ \frac{{\partial }^{2}𝑓}{\partial 𝑦\partial 𝑥} & \frac{{\partial }^{2}𝑓}{\partial {𝑦}^{2}}\end{pmatrix}

(3.35)

Vandermonde, because a power in a cell is not the same as a superscript on the whole matrix:

\begin{pmatrix}1 & {𝑥}_{1} & {𝑥}_{1}^{2} \\ 1 & {𝑥}_{2} & {𝑥}_{2}^{2} \\ 1 & {𝑥}_{3} & {𝑥}_{3}^{2}\end{pmatrix}

(3.36)

Pauli matrices. If $𝑖$ becomes italic garbage, I will notice.

{\sigma }_{1}=\begin{pmatrix}0 & 1 \\ 1 & 0\end{pmatrix},\quad {\sigma }_{2}=\begin{pmatrix}0 & -𝑖 \\ 𝑖 & 0\end{pmatrix},\quad {\sigma }_{3}=\begin{pmatrix}1 & 0 \\ 0 & -1\end{pmatrix}

(3.37)

A block matrix, letters standing for blocks I have not opened:

\begin{bmatrix}𝐴 & 𝐵 \\ 𝐶 & 𝐷\end{bmatrix}\begin{bmatrix}𝑥 \\ 𝑦\end{bmatrix}=\begin{bmatrix}𝐴𝑥+𝐵𝑦 \\ 𝐶𝑥+𝐷𝑦\end{bmatrix}

(3.38)

The characteristic polynomial as a determinant, $\lambda$ in the diagonal cells:

\det (𝐴-\lambda 𝐼)=\begin{vmatrix}𝑎-\lambda & 𝑏 \\ 𝑐 & 𝑑-\lambda \end{vmatrix}=(𝑎-\lambda )(𝑑-\lambda )-𝑏𝑐

(3.39)

Cases are an array with a brace. Absolute value is the smallest example I already used; here is a flux that changes sign.

𝑓(𝑥)=\begin{cases}{𝑥}^{2}\, & \text{if}\, 𝑥\geq 0 \\ -{𝑥}^{2}\, & \text{if}\, 𝑥<0\end{cases}

(3.40)

Three branches. Two is not a gauntlet.

\text{sgn}(𝑥)=\begin{cases}1\, & \text{if}\, 𝑥>0 \\ 0\, & \text{if}\, 𝑥=0 \\ -1\, & \text{if}\, 𝑥<0\end{cases}

(3.41)

Heaviside, written as a step I will integrate against:

𝐻(𝑥)=\begin{cases}0\, & \text{if}\, 𝑥<0 \\ 1\, & \text{if}\, 𝑥\geq 0\end{cases}

(3.42)

A piecewise ODE, because a system can change formula across a threshold:

{𝑥}^{\prime }=\begin{cases}-𝑥\, & \text{if}\, |𝑥|>1 \\ -{𝑥}^{3}\, & \text{if}\, |𝑥|\leq 1\end{cases}

(3.43)