In mathematics, a square matrix is a matrix with the same number of rows and columns. An n-by-n matrix is known as a square matrix of order ${\displaystyle n}$. Any two square matrices of the same order can be added and multiplied.

Square matrices are often used to represent simple linear transformations, such as shearing or rotation. For example, if ${\displaystyle R}$ is a square matrix representing a rotation (rotation matrix) and ${\displaystyle \mathbf {v} }$ is a column vector describing the position of a point in space, the product ${\displaystyle R\mathbf {v} }$ yields another column vector describing the position of that point after that rotation. If ${\displaystyle \mathbf {v} }$ is a row vector, the same transformation can be obtained using ${\displaystyle \mathbf {v} R^{\mathsf {T))}$, where ${\displaystyle R^{\mathsf {T))}$ is the transpose of ${\displaystyle R}$.

Main diagonal

The entries ${\displaystyle a_{ii))$ (i = 1, ..., n) form the main diagonal of a square matrix. They lie on the imaginary line which runs from the top left corner to the bottom right corner of the matrix. For instance, the main diagonal of the 4×4 matrix above contains the elements a₁₁ = 9, a₂₂ = 11, a₃₃ = 4, a₄₄ = 10.

The diagonal of a square matrix from the top right to the bottom left corner is called antidiagonal or counterdiagonal.

Special kinds

Name	Example with n = 3
Diagonal matrix	${\displaystyle {\begin{bmatrix}a_{11}&0&0\\0&a_{22}&0\\0&0&a_{33}\end{bmatrix))}$
Lower triangular matrix	${\displaystyle {\begin{bmatrix}a_{11}&0&0\\a_{21}&a_{22}&0\\a_{31}&a_{32}&a_{33}\end{bmatrix))}$
Upper triangular matrix	${\displaystyle {\begin{bmatrix}a_{11}&a_{12}&a_{13}\\0&a_{22}&a_{23}\\0&0&a_{33}\end{bmatrix))}$

Diagonal or triangular matrix

If all entries outside the main diagonal are zero, ${\displaystyle A}$ is called a diagonal matrix. If only all entries above (or below) the main diagonal are zero, ${\displaystyle A}$ is called an upper (or lower) triangular matrix.

Identity matrix

The identity matrix ${\displaystyle I_{n))$ of size ${\displaystyle n}$ is the ${\displaystyle n\times n}$ matrix in which all the elements on the main diagonal are equal to 1 and all other elements are equal to 0, e.g.

$${\displaystyle I_{1}={\begin{bmatrix}1\end{bmatrix)),\ I_{2}={\begin{bmatrix}1&0\\0&1\end{bmatrix)),\ \ldots ,\ I_{n}={\begin{bmatrix}1&0&\cdots &0\\0&1&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &1\end{bmatrix)).}$$

${\displaystyle n}$,

$${\displaystyle AI_{n}=I_{m}A=A}$$

${\displaystyle A}$.

Invertible matrix and its inverse

A square matrix ${\displaystyle A}$ is called invertible or non-singular if there exists a matrix ${\displaystyle B}$ such that^[1][2]

$${\displaystyle AB=BA=I_{n}.}$$

${\displaystyle B}$

${\displaystyle A}$,

${\displaystyle A^{-1))$.

Symmetric or skew-symmetric matrix

A square matrix ${\displaystyle A}$ that is equal to its transpose, i.e., ${\displaystyle A^{\mathsf {T))=A}$, is a symmetric matrix. If instead ${\displaystyle A^{\mathsf {T))=-A}$, then ${\displaystyle A}$ is called a skew-symmetric matrix.

For a complex square matrix ${\displaystyle A}$, often the appropriate analogue of the transpose is the conjugate transpose ${\displaystyle A^{*))$, defined as the transpose of the complex conjugate of ${\displaystyle A}$. A complex square matrix ${\displaystyle A}$ satisfying ${\displaystyle A^{*}=A}$ is called a Hermitian matrix. If instead ${\displaystyle A^{*}=-A}$, then ${\displaystyle A}$ is called a skew-Hermitian matrix.

By the spectral theorem, real symmetric (or complex Hermitian) matrices have an orthogonal (or unitary) eigenbasis; i.e., every vector is expressible as a linear combination of eigenvectors. In both cases, all eigenvalues are real.^[3]

Definite matrix

Positive definite	Indefinite
${\displaystyle {\begin{bmatrix}1/4&0\\0&1\\\end{bmatrix))}$	${\displaystyle {\begin{bmatrix}1/4&0\\0&-1/4\end{bmatrix))}$
Q(x,y) = 1/4 x2 + y2	Q(x,y) = 1/4 x2 − 1/4 y2
Points such that Q(x, y) = 1 (Ellipse).	Points such that Q(x, y) = 1 (Hyperbola).

A symmetric n×n-matrix is called positive-definite (respectively negative-definite; indefinite), if for all nonzero vectors ${\displaystyle x\in \mathbb {R} ^{n))$ the associated quadratic form given by

$${\displaystyle Q(\mathbf {x} )=\mathbf {x} ^{\mathsf {T))A\mathbf {x} }$$

A symmetric matrix is positive-definite if and only if all its eigenvalues are positive.^[5] The table at the right shows two possibilities for 2×2 matrices.

Allowing as input two different vectors instead yields the bilinear form associated to A:^[6]

$${\displaystyle B_{A}(\mathbf {x} ,\mathbf {y} )=\mathbf {x} ^{\mathsf {T))A\mathbf {y} .}$$

Orthogonal matrix

An orthogonal matrix is a square matrix with real entries whose columns and rows are orthogonal unit vectors (i.e., orthonormal vectors). Equivalently, a matrix A is orthogonal if its transpose is equal to its inverse:

$${\displaystyle A^{\textsf {T))=A^{-1},}$$

$${\displaystyle A^{\textsf {T))A=AA^{\textsf {T))=I,}$$

An orthogonal matrix A is necessarily invertible (with inverse A⁻¹ = A^T), unitary (A⁻¹ = A*), and normal (A*A = AA*). The determinant of any orthogonal matrix is either +1 or −1. The special orthogonal group ${\displaystyle \operatorname {SO} (n)}$ consists of the n × n orthogonal matrices with determinant +1.

The complex analogue of an orthogonal matrix is a unitary matrix.

Normal matrix

A real or complex square matrix ${\displaystyle A}$ is called normal if ${\displaystyle A^{*}A=AA^{*))$. If a real square matrix is symmetric, skew-symmetric, or orthogonal, then it is normal. If a complex square matrix is Hermitian, skew-Hermitian, or unitary, then it is normal. Normal matrices are of interest mainly because they include the types of matrices just listed and form the broadest class of matrices for which the spectral theorem holds.^[7]

Operations

Trace

The trace, tr(A) of a square matrix A is the sum of its diagonal entries. While matrix multiplication is not commutative, the trace of the product of two matrices is independent of the order of the factors:

$${\displaystyle \operatorname {tr} (AB)=\operatorname {tr} (BA).}$$

$${\displaystyle \operatorname {tr} (AB)=\sum _{i=1}^{m}\sum _{j=1}^{n}A_{ij}B_{ji}=\operatorname {tr} (BA).}$$

$${\displaystyle \operatorname {tr} (A)=\operatorname {tr} (A^{\mathrm {T} }).}$$

Determinant

The determinant ${\displaystyle \det(A)}$ or ${\displaystyle |A|}$ of a square matrix ${\displaystyle A}$ is a number encoding certain properties of the matrix. A matrix is invertible if and only if its determinant is nonzero. Its absolute value equals the area (in ${\displaystyle \mathbb {R} ^{2))$) or volume (in ${\displaystyle \mathbb {R} ^{3))$) of the image of the unit square (or cube), while its sign corresponds to the orientation of the corresponding linear map: the determinant is positive if and only if the orientation is preserved.

The determinant of 2×2 matrices is given by

$${\displaystyle \det {\begin{bmatrix}a&b\\c&d\end{bmatrix))=ad-bc.}$$

The determinant of a product of square matrices equals the product of their determinants:^[9]

$${\displaystyle \det(AB)=\det(A)\cdot \det(B)}$$

Eigenvalues and eigenvectors

A number λ and a non-zero vector ${\displaystyle \mathbf {v} }$ satisfying

$${\displaystyle A\mathbf {v} =\lambda \mathbf {v} }$$

${\displaystyle A}$,

$${\displaystyle \det(A-\lambda I)=0.}$$