Matrix analysis

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.

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).

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.

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.

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