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Matrix Power

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

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Matrix Power -

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Matrix Power

Matrix Power


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Answers to Questions (FAQ)

What is a matrix power? (Definition)

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.

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

How to calculate the matrix power 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.

Why diagonalizing a matrix before calculating its power?

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

How to compute a negative power of a 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: $$ \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} ^{-2} = \left( \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} ^{-1} \right)^2 $$

How to compute a matrix root?

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

How to compute a noninteger power of a matrix?

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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