# Matrix Types Cheat Sheet

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:

$$
A_{i,j}=0\Leftarrow i\ne j
$$

A **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:

$$
I_{i,j}=\begin{cases}1&\text{if }i=j\\0&\text{otherwise}\end{cases}
$$

A **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 *unitary*, *Hermitian*, and *skew-Hermitian*) commutes with its conjugate transpose:

$$\boldsymbol{A}^*\boldsymbol{A}=\boldsymbol{A}\boldsymbol{A}^*$$

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:

$$\boldsymbol{A}=\boldsymbol{A}^\mathsf{T}$$

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

$$\boldsymbol{A}=-\boldsymbol{A}^\mathsf{T}$$

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

$$\boldsymbol{H}=\boldsymbol{H}^*$$

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

$$\boldsymbol{H}=-\boldsymbol{H}^*$$

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

$$\boldsymbol{A}\boldsymbol{B}=\boldsymbol{B}\boldsymbol{A}=\boldsymbol{I}_n$$

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

A **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:

$$\boldsymbol{A}^\mathsf{T}=\boldsymbol{A}^{-1}\iff\boldsymbol{A}^\mathsf{T}\boldsymbol{A}=\boldsymbol{A}\boldsymbol{A}^\mathsf{T}=\boldsymbol{I}$$

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:

$$\boldsymbol{A}\boldsymbol{A}=\boldsymbol{A}$$

A *square* matrix $\boldsymbol{A}$ is **diagonalizable** (or **nondefective**; wiki) 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.

A **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*.

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

A **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:

$$A_{i,j}=\begin{cases}

1&\text{if }i=j\\

A_{i,j}&\text{if }i<j\land j=j’\\

0&\text{otherwise}

\end{cases}$$

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._