Matrix

2 min read#linear-algebra

A rectangular grid of numbers that doubles as a recipe for transforming space.

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Matrix

A matrix is a rectangular grid of numbers, written with rows and columns:

A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}.

On the surface it is just bookkeeping — a table of coefficients. Its real meaning is that a matrix is a Linear Transformation: a rule for moving every Vector in space at once. Feeding a vector \vec v = (x, y) to the matrix produces a new vector A\vec v, and the whole power of the subject comes from reading the grid of numbers and the geometric warp it encodes as two faces of one thing.

How a matrix eats a vector

Matrix–vector multiplication is a weighted sum of the matrix's columns:

A\vec v = \begin{bmatrix} a & b \\ c & d \end{bmatrix}\begin{bmatrix} x \\ y \end{bmatrix} = x\begin{bmatrix} a \\ c \end{bmatrix} + y\begin{bmatrix} b \\ d \end{bmatrix}.
(1)

This is the key to seeing a matrix. The first column is where the matrix sends the basis vector (1,0); the second column is where it sends (0,1). Every other vector is carried along as the same linear combination of those two landing spots. So you can read a 2\times2 matrix at a glance: its columns are the images of the coordinate axes.

Special matrices

  • The identity \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} leaves the axes — and therefore every vector — exactly where they are.
  • A diagonal matrix \begin{bmatrix} s & 0 \\ 0 & t \end{bmatrix} stretches the x-axis by s and the y-axis by t.
  • A rotation \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix} turns every vector by \theta without changing its length.

See also