Editing Matrix multiplication (section)

==Square matrices==
Let us denote <math>\mathcal M_n(R)</math> the set of {{math|''n''×''n''}} [[square matrices]] with entries in a [[ring (mathematics)|ring]] {{mvar|R}}, which, in practice, is often a [[field (mathematics)|field]].

In <math>\mathcal M_n(R)</math>, the product is defined for every pair of matrices. This makes <math>\mathcal M_n(R)</math> a [[ring (mathematics)|ring]], which has the [[identity matrix]] {{math|'''I'''}} as [[identity element]] (the matrix whose diagonal entries are equal to 1 and all other entries are 0). This ring is also an [[associative algebra|associative {{mvar|R}}-algebra]].

If {{math|''n'' > 1}}, many matrices do not have a [[multiplicative inverse]]. For example, a matrix such that all entries of a row (or a column) are 0 does not have an inverse. If it exists, the inverse of a matrix {{math|'''A'''}} is denoted {{math|'''A'''<sup>−1</sup>}}, and, thus verifies
:<math> \mathbf{A}\mathbf{A}^{-1} = \mathbf{A}^{-1}\mathbf{A} = \mathbf{I}. </math>
A matrix that has an inverse is an [[invertible matrix]]. Otherwise, it is a [[singular matrix]].

A product of matrices is invertible if and only if each factor is invertible. In this case, one has

:<math>(\mathbf{A}\mathbf{B})^{-1} = \mathbf{B}^{-1}\mathbf{A}^{-1}.</math>

When {{mvar|R}} is [[commutative ring|commutative]], and, in particular, when it is a field, the [[determinant]] of a product is the product of the determinants. As determinants are scalars, and scalars commute, one has thus 
:<math> \det(\mathbf{AB}) = \det(\mathbf{BA}) =\det(\mathbf{A})\det(\mathbf{B}). </math>

The other matrix [[invariant (mathematics)|invariants]] do not behave as well with products. Nevertheless, if {{mvar|R}} is commutative, {{math|'''AB'''}} and {{math|'''BA'''}} have the same [[Trace (linear algebra)|trace]], the same [[characteristic polynomial]], and the same [[eigenvalues]] with the same multiplicities. However, the [[eigenvector]]s are generally different if {{math|'''AB''' ≠ '''BA'''}}.

===Powers of a matrix===
One may raise a square matrix to any [[exponentiation|nonnegative integer power]] multiplying it by itself repeatedly in the same way as for ordinary numbers. That is,
:<math>\mathbf{A}^0 = \mathbf{I},</math>
:<math>\mathbf{A}^1 = \mathbf{A},</math>
:<math>\mathbf{A}^k = \underbrace{\mathbf{A}\mathbf{A}\cdots\mathbf{A}}_{k\text{ times}}.</math>

Computing the {{mvar|k}}th power of a matrix needs {{math|''k'' – 1}} times the time of a single matrix multiplication, if it is done with the trivial algorithm (repeated multiplication). As this may be very time consuming, one generally prefers using [[exponentiation by squaring]], which requires less than {{math|2 log<sub>2</sub> ''k''}} matrix multiplications, and is therefore much more efficient.

An easy case for exponentiation is that of a [[diagonal matrix]]. Since the product of diagonal matrices amounts to simply multiplying corresponding diagonal elements together, the {{mvar|k}}th power of a diagonal matrix is obtained by raising the entries to the power {{mvar|k}}:
:<math>
  \begin{bmatrix}
    a_{11} & 0      & \cdots & 0 \\
    0      & a_{22} & \cdots & 0 \\
    \vdots & \vdots & \ddots & \vdots \\
    0     & 0       & \cdots & a_{nn}
  \end{bmatrix}^k =
\begin{bmatrix}
    a_{11}^k & 0        & \cdots & 0 \\
    0        & a_{22}^k & \cdots & 0 \\
    \vdots   & \vdots   & \ddots & \vdots \\
    0        & 0        & \cdots & a_{nn}^k
  \end{bmatrix}.
</math>