Matrix Power

The exponentiation of matrix $ M $ to the power $ n $ ($ n \neq 0 $) is written $ M^n $ and is defined as the multiplication">matrix product (the multiplication) of $ M $ by itself $ n $ times.

$$ M^n = \underbrace{M \cdot M \cdot \ldots \cdot M}_{n} $$

Taking $ M $ a square matrix of size $ m $ ($ m $ rows and $ m $ columns).

The size of the resulting matrix is identical to the original matrix M; i.e. $ m $ lines and $ m $ columns.

If the matrix is diagonalizable, then its diagonalization greatly simplifies the power calculations because it applies mainly on the diagonal of the matrix.

Calculating $ M^{-n} $ is equivalent to $ M^{-1 \times n} $. Thus, calculate the inverse of the matrix and then perform with it an exponentiation to the power $ n $.

The calculation of $ M^{1/n} $ is equivalent to the $ n $ -th root.

The exponentiation $ n $ (with $ n $ a nonzero real number) of an invertible square matrix $ M $ can be defined by $ M^n = \exp(n \log{M}) $ and therefore the power of the matrix can be calculated with a decimal number as the exponent. In this case, the logarithm of a matrix is defined with the eigenvectors $ V $ of $ M $ such that $ \log{M} = V . \log{ V^{-1} . A . V } . V^{-1} $ and the exponential of a matrix is can be calculated using an integer series $ e^M = \sum_{k=0}^{\infty} \frac{1}{k!} M^k $.