Editing Matrix multiplication (section)

===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>