Timo Denk's Blog

Matrix Types Cheat Sheet

· Timo Denk

In the field of linear algebra there are variety of different matrix types. Each has its own definition and relevance. I had trouble finding a good overview online and thought I’d compile a list myself: This article lists a selection of matrix types as well as their definition, mostly based on the corresponding Wikipedia articles. Generally, I recommend The Matrix Cookbook for concise facts about matrices and this figure on $n\times n$ matrices.

A matrix is diagonal (wiki) if all entries outside the main diagonal are zero:
$$ A_{i,j}=0\Leftarrow i\ne j $$

square matrix (wiki) is a matrix with the same number of rows and columns, e.g. $n\times n$.

An identity matrix $\boldsymbol{I}_n$ (wiki) is a diagonal square matrix whose entries on the main diagonal are one:
$$ I_{i,j}=\begin{cases}1&\text{if }i=j\\0&\text{otherwise}\end{cases} $$

zero matrix $\boldsymbol{0}_{n,m}$ (wiki) is a $n\times m$ matrix whose entries are zero (respectively, a one matrix $\boldsymbol{1}_{n,m}$ has only one entries):
$$ 0_{i,j}=0 $$

A normal matrix (wiki; always unitaryHermitian, and skew-Hermitian) commutes with its conjugate transpose:

An upper triangular matrix (wiki) has only zero entries below its main diagonal:
$$A_{i,j}=0\Leftarrow i\lt j$$

A lower triangular matrix (wiki) has only zero entries above its main diagonal:
$$A_{i,j}=0\Leftarrow i\gt j$$

A symmetric matrix (wiki) is equal to its transpose:

A skew-symmetric matrix (wiki) is equal to the negative of its transpose:

Hermitian (or self-adjoint) matrix (wiki) is a complex square matrix that is equal to its own conjugate transpose:

skew-Hermitian (or antihermitian) matrix (wiki) is a complex square matrix whose conjugate transpose is the negative of the original matrix:

For an invertible (also nonsingular or nondegeneratesquare matrix $\boldsymbol{A}$ (wiki) there exists a matrix $\boldsymbol{B}$ which is inverse to $\boldsymbol{A}$:

A singular (or degenerate) matrix (wiki) is not invertible.

cofactor matrix $\boldsymbol{C}$ (wiki; also matrix of cofactors or comatrix) of a square matrix $\boldsymbol{A}$ is defined such that the inverse of $\boldsymbol{A}$ is the transpose of the cofactor matrix times the reciprocal of the determinant of $\boldsymbol{A}$:
$$\boldsymbol{A}^{-1} = \frac{1}{\operatorname{det}(\boldsymbol{A})} \boldsymbol{C}^\mathsf{T}$$

The transpose of an orthogonal matrix (wiki) is equal to its inverse:

A matrix is unitary (wiki) if its conjugate transpose $\boldsymbol{U}^*$ is also its inverse: $$\boldsymbol{U}^*\boldsymbol{U}=\boldsymbol{U}\boldsymbol{U}^*=\boldsymbol{I}$$

A _symmetric _square real matrix $\boldsymbol{A}$ is positive-definite (wiki), if for every non-zero column vector $\boldsymbol{z}$, $$\boldsymbol{z}^\intercal\boldsymbol{A}\boldsymbol{z}\gt0$$ holds. For negative-definite matrices, $\boldsymbol{z}^\intercal\boldsymbol{A}\boldsymbol{z}\lt0$. In the complex case, the Hermitian matrix $\boldsymbol{H}$ satisfies $\boldsymbol{z}^*\boldsymbol{H}\boldsymbol{z}\gt0$ (or $\boldsymbol{z}^*\boldsymbol{H}\boldsymbol{z}\lt0$ respectively).

A positive-semidefinite (wiki; or negative-semidefinite) matrix is defined similarly to positive-definite and negative-definite matrices, with the difference that the greater than and less than comparisons are relaxed to allow for zero scalars as well.

An idempotent matrix (wiki) is a square matrix which, when multiplied by itself, yields itself:

A square matrix $\boldsymbol{A}$ is diagonalizable (or nondefectivewiki) if it there exists a matrix $\boldsymbol{P}$ and its inverse $\boldsymbol{P}^{-1}$ such that $$\boldsymbol{P}^{-1}\boldsymbol{A}\boldsymbol{P}$$is a diagonal matrix.

permutation matrix (wiki) is a square binary matrix that has exactly one entry of one in each row and each column and zeros elsewhere. It is orthogonal.

submatrix (wiki) of another matrix is obtained by deleting any collection of rows and/or columns from it.

Frobenius matrix (wiki) is a square matrix with the properties (1) all entries on the main diagonal are one, (2) the entries below the main diagonal of at most one column $j’$ are arbitrary, and (3) every other entry is zero:
1&\text{if }i=j\\
A_{i,j}&\text{if }i<j\land j=j’\\

A Toeplitz matrix (wiki) is a matrix in which each descending diagonal from left to right is constant.

Post title photo by Tadas Sar on Unsplash. With some imagination, there are lots of square matrices composing the building’s wall._