In mathematics, particularly in linear algebra and applications, matrix analysis is the study of matrices and their algebraic properties.^[1] Some particular topics out of many include; operations defined on matrices (such as matrix addition, matrix multiplication and operations derived from these), functions of matrices (such as matrix exponentiation and matrix logarithm, and even sines and cosines etc. of matrices), and the eigenvalues of matrices (eigendecomposition of a matrix, eigenvalue perturbation theory).^[2]

Matrix spaces

The set of all m × n matrices over a field F denoted in this article M_mn(F) form a vector space. Examples of F include the set of rational numbers ${\displaystyle \mathbb {Q} }$, the real numbers ${\displaystyle \mathbb {R} }$, and set of complex numbers ${\displaystyle \mathbb {C} }$. The spaces M_mn(F) and M_pq(F) are different spaces if m and p are unequal, and if n and q are unequal; for instance M₃₂(F) ≠ M₂₃(F). Two m × n matrices A and B in M_mn(F) can be added together to form another matrix in the space M_mn(F):

${\displaystyle \mathbf {A} ,\mathbf {B} \in M_{mn}(F)\,,\quad \mathbf {A} +\mathbf {B} \in M_{mn}(F)}$

and multiplied by a α in F, to obtain another matrix in M_mn(F):

${\displaystyle \alpha \in F\,,\quad \alpha \mathbf {A} \in M_{mn}(F)}$

Combining these two properties, a linear combination of matrices A and B are in M_mn(F) is another matrix in M_mn(F):

${\displaystyle \alpha \mathbf {A} +\beta \mathbf {B} \in M_{mn}(F)}$

where α and β are numbers in F.

Any matrix can be expressed as a linear combination of basis matrices, which play the role of the basis vectors for the matrix space. For example, for the set of 2 × 2 matrices over the field of real numbers, ${\displaystyle M_{22}(\mathbb {R} )}$, one legitimate basis set of matrices is:

${\displaystyle {\begin{pmatrix}1&0\\0&0\end{pmatrix}}\,,\quad {\begin{pmatrix}0&1\\0&0\end{pmatrix}}\,,\quad {\begin{pmatrix}0&0\\1&0\end{pmatrix}}\,,\quad {\begin{pmatrix}0&0\\0&1\end{pmatrix}}\,,}$

because any 2 × 2 matrix can be expressed as:

${\displaystyle {\begin{pmatrix}a&b\\c&d\end{pmatrix}}=a{\begin{pmatrix}1&0\\0&0\end{pmatrix}}+b{\begin{pmatrix}0&1\\0&0\end{pmatrix}}+c{\begin{pmatrix}0&0\\1&0\end{pmatrix}}+d{\begin{pmatrix}0&0\\0&1\end{pmatrix}}\,,}$

where a, b, c,d are all real numbers. This idea applies to other fields and matrices of higher dimensions.

Determinants

The determinant of a square matrix is an important property. The determinant indicates if a matrix is invertible (i.e. the inverse of a matrix exists when the determinant is nonzero). Determinants are used for finding eigenvalues of matrices (see below), and for solving a system of linear equations (see Cramer's rule).

Eigenvalues and eigenvectors of matrices

Matrix similarity

Two n × n matrices A and B are similar if they are related by a similarity transformation:

${\displaystyle \mathbf {B} =\mathbf {P} \mathbf {A} \mathbf {P} ^{-1}}$

The matrix P is called a similarity matrix, and is necessarily invertible.

Canonical forms

Triangular factorization

Matrix norms

Since matrices form vector spaces, one can form axioms (analogous to those of vectors) to define a "size" of a particular matrix. The norm of a matrix is a positive real number.

Positive definite and semidefinite matrices

Functions

Matrix elements are not restricted to constant numbers, they can be mathematical variables.

See also

Footnotes

References