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

\( M \) is a square matrix of site \( m \) (\( m \) rows and \( m \) columns). The calculation of the \( n \)th power of the matrix \( M \) is denoted by \( M^n \) (\( M \) power \( n \)) and consists of multiplying the matrix \( n \) times by itself.

Example: Power of a 2x2 matrix: $$ \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} ^2 = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \times \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} = \begin{bmatrix} 7 & 10 \\ 15 & 22 \end{bmatrix} $$

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

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