Matrix

Matrices are foundational to linear algebra. Their sizes are expressed as $n\times m$, row by column. They take inputs from their domain ${\mathbb{R}}^m$ and output into their codomain ${\mathbb{R}}^n$.

$$T(\vec{x})=A\vec{x}:{\mathbb{R}}^m\to {\mathbb{R}}^n$$

Basic terminology

• We describe matrices with the same number of rows and columns as square matrices.
• Diagonal matrices only have elements along their diagonals; they’re always square.
• Upper and lower triangular matrices only have elements in their top-right or bottom-left halves; these include the diagonals.

The alphabet

A brief list based on what I’ve seen so far:¹

• $A$: any matrix
• $B$: basis matrix
• $D$: diagonal matrix
• $I_n$: identity matrix, an $n\times n$ matrix with only ones along its diagonal and zeroes otherwise²
• $L$: lower triangular matrix
• $Q$: orthogonal matrix
• $R$: upper triangular matrix
• $T$: linear transformation
• $\Sigma$: singular value matrix

We also define $A^T$ as the transpose of $A$, i.e., where the rows and columns are swapped.

Theorems

• $T$ is a linear transformation if and only if:
  • $T(\vec{x}+\vec{y})=T\vec{x}+T\vec{y}$
  • $T(k\vec{x})=k(T\vec{x})$

Related concepts

• Matrix multiplication
• Inverse relation
• Matrix similarity

Computer-assisted

MATLAB often has some useful tools available for matrix manipulation. To represent some $3\times 3$ matrix, we can use spaces to separate entries in the same row, and semi-colons to separate rows.

[a b c; d e f; g h i]

Footnotes

1. Adapted from the Introduction to Linear Algebra book by Gilbert Strang. ↩

2. In MATLAB, $I_n$ can be generated with A = eye(n). ↩