Matrices

Concept

A vector is an ordered collection of components. A matrix is an ordered collection of entries arranged in rows and columns. It can record relationships, combine values, or act on a vector.

Equation

$A=[a_{11}{a}_{12};a_{21}{a}_{22}]$

Addition

Matrices can be added only when they have the same dimensions. That makes every entry in one matrix correspond to exactly one entry in the other.

Scalar multiplication

Matrix × vector

This is the central operation: each row of A takes a dot product with the input vector and produces one output component. Select a row calculation to trace it.

Matrix × matrix

For $A:m\times n$ and $B:n\times p$, the matching inner dimensions allow multiplication, and the outer dimensions give $AB:m\times p$.

Observe

The identity matrix leaves a vector unchanged: $Ix=x$, and it satisfies $AI=IA=A$. The zero matrix sends every vector to zero: $0x=0$. Try the Identity and Swap presets to see both ideas.

Explain

Matrix multiplication is not just a rule for filling cells. It describes composition: first use B to combine values, then use A to combine the result. Reversing that order can produce a different answer.

Computing, data & machine learning

Datasets use rows for observations and columns for features. Images are grids of pixel values. Graphics uses matrices to encode transformations. In machine learning, a layer is often summarized as $y=Wx+b$: $x$ is an input vector, $W$ is a weight matrix, $b$ is a bias vector, and $y$ is an output vector.

Matrix transformations

What happens to the whole coordinate plane when every vector is multiplied by A?