A matrix is a 2d table of number. It has a size called the dimension. An example
of a matrix of dimension 3 × 3 3 \times 3 3 × 3 ( 1 2 3 4 5 6 7 8 9 ) \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6
\\ 7 & 8 & 9 \end{pmatrix} 1 4 7 2 5 8 3 6 9
You can add and multiply matrices together. Addition works like you'd think but
not multiplication !
[ a 11 a 12 ⋯ a 1 n a 21 a 22 ⋯ a 2 n ⋮ ⋮ ⋱ ⋮ a m 1 a m 2 ⋯ a m n ] × [ b 11 b 12 ⋯ b 1 p b 21 b 22 ⋯ b 2 p ⋮ ⋮ ⋱ ⋮ b n 1 b n 2 ⋯ b n p ] = [ c 11 c 12 ⋯ c 1 p c 21 c 22 ⋯ c 2 p ⋮ ⋮ ⋱ ⋮ c m 1 c m 2 ⋯ c m p ] \begin{bmatrix}
a_{11} & a_{12} & \cdots & a_{1n}\\
a_{21} & a_{22} & \cdots & a_{2n}\\
\vdots & \vdots & \ddots & \vdots\\
a_{m1} & a_{m2} & \cdots & a_{mn}
\end{bmatrix}
\times
\begin{bmatrix}
b_{11} & b_{12} & \cdots & b_{1p}\\
b_{21} & b_{22} & \cdots & b_{2p}\\
\vdots & \vdots & \ddots & \vdots\\
b_{n1} & b_{n2} & \cdots & b_{np}
\end{bmatrix}
=
\begin{bmatrix}
c_{11} & c_{12} & \cdots & c_{1p}\\
c_{21} & c_{22} & \cdots & c_{2p}\\
\vdots & \vdots & \ddots & \vdots\\
c_{m1} & c_{m2} & \cdots & c_{mp}
\end{bmatrix} a 11 a 21 ⋮ a m 1 a 12 a 22 ⋮ a m 2 ⋯ ⋯ ⋱ ⋯ a 1 n a 2 n ⋮ a mn × b 11 b 21 ⋮ b n 1 b 12 b 22 ⋮ b n 2 ⋯ ⋯ ⋱ ⋯ b 1 p b 2 p ⋮ b n p = c 11 c 21 ⋮ c m 1 c 12 c 22 ⋮ c m 2 ⋯ ⋯ ⋱ ⋯ c 1 p c 2 p ⋮ c m p With: c ∗ i j = a ∗ i 1 b ∗ 1 j + a ∗ i 2 b ∗ 2 j + ⋯ + a ∗ i n + b ∗ n j = ∑ ∗ k = 1 n a ∗ i k b ∗ k j c*{ij}= a*{i1} b*{1j} + a*{i2} b*{2j} +\cdots+ a*{in} + b*{nj} =
\sum*{k=1}^n a*{ik}b*{kj} c ∗ ij = a ∗ i 1 b ∗ 1 j + a ∗ i 2 b ∗ 2 j + ⋯ + a ∗ in + b ∗ nj = ∑ ∗ k = 1 n a ∗ ik b ∗ kj i , j i,j i , j
Matrix are a complex topic and this is not a full linear algebra course. The
most important thing to understand is that matrices allow you to represent
functions that act upon vectors in a "linear".