Search for a tool
Matrix Power

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

Results

Matrix Power -

Tag(s) : Matrix

Share dCode and you

dCode is free and its tools are a valuable help in games, maths, geocaching, puzzles and problems to solve every day!
A suggestion ? a feedback ? a bug ? an idea ? Write to dCode!

Please, check our community Discord for help requests!

Thanks to your feedback and relevant comments, dCode has developped the best Matrix Power tool, so feel free to write! Thank you !

# Matrix Power

## Matrix Power

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

### How to calculate the matrix power n?

$M$ 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: (squared) $$\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} ^2 = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \times \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} = \begin{bmatrix} 7 & 10 \\ 15 & 22 \end{bmatrix}$$

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

## Source code

dCode retains ownership of the online 'Matrix Power' tool source code. Except explicit open source licence (indicated CC / Creative Commons / free), any algorithm, applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, translator), or any function (convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (PHP, Java, C#, Python, Javascript, Matlab, etc.) no data, script or API access will be for free, same for Matrix Power download for offline use on PC, tablet, iPhone or Android !

## Need Help ?

Please, check our community Discord for help requests!