Tool to calculate matrix exponential in algebra. Matrix power consists in exponentiation of the matrix (multiplication by itself).

Matrix Power - dCode

Tag(s) : Matrix

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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).

__Example:__ Power of a 2x2 matrix squared (raised to power 2) $$ \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} ^2 = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \times \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} = \begin{bmatrix} 7 & 10 \\ 15 & 22 \end{bmatrix} $$

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

Calculating matrix power only works for square matrices (2x2, 3x3, 4x4, 5x5, etc. due to constraints with multiplication">matrix products) and is used for some matrices such as stochastic matrices.

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 $.

__Example:__ $$ \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} ^{-2} = \left( \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} ^{-1} \right)^2 $$

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 $.

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*Matrix Power* on dCode.fr [online website], retrieved on 2024-09-13,

power,exponent,square,matrix

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