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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

## Matrix Power

### 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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