## Power of a Matrix

A repeat multiplication of a square matrix with itself times produces matrix power , thus

When the power is large, computation of matrix multiplication takes a long time and a more efficient method is to transform matrix into a similar diagonal matrix using similarity transformation . Matrix (is called modal matrix) is formed by horizontal concatenation of the linearly independent eigenvectors . Matrix can be obtained back from diagonal matrix using . The computation of matrix power is simplified by taking the power to the scalar diagonal elements. Then multiply the diagonal power with the modal matrix and its inverse .

When the power is not integer the computation of matrix power is possible only through similarity transformation into diagonal matrix . The diagonalization can take place only if the modal matrix has an inverse matrix .

**
Example:
**

From matrix
we can obtain diagonal matrix
. Through modal matrix (augmented eigenvectors)
. The inverse modal matrix is
.

Then, matrix power can be computed through repeated matrix multiplication 4 times
. The same result can be obtained using diagonal matrix
. Then we multiply with modal matrix and its inverse, require only three matrix multiplications
.

Clearly this procedure is only more efficient than matrix multiplication when the power
is large.

However, we have something else that matrix multiplication cannot perform: when matrix power is not integer.

**
Example:
**

Using the matrices in previous example, find

In this example, matrix multiplication cannot solve our problem.

Solution: We can take the exponent to the diagonal elements of the diagonal matrix as in scalar
. We multiply back to obtain the matrix power
.

**
See also
**
:
Matrix Eigen Value & Eigen Vector
,
Similarity and Matrix Diagonalization

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