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 .
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.
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 .