By Kardi Teknomo, PhD .
LinearAlgebra

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Power of a Matrix

A repeat multiplication of a square matrix Matrix Power and Matrix Exponent with itself Matrix Power and Matrix Exponent times produces matrix power Matrix Power and Matrix Exponent , thus Matrix Power and Matrix Exponent

When the power Matrix Power and Matrix Exponent is large, computation of matrix multiplication takes a long time and a more efficient method is to transform matrix Matrix Power and Matrix Exponent into a similar diagonal matrix Matrix Power and Matrix Exponent using similarity transformation Matrix Power and Matrix Exponent . Matrix Matrix Power and Matrix Exponent (is called modal matrix) is formed by horizontal concatenation of the Matrix Power and Matrix Exponent linearly independent eigenvectors Matrix Power and Matrix Exponent . Matrix Matrix Power and Matrix Exponent can be obtained back from diagonal matrix using Matrix Power and Matrix Exponent . 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 Matrix Power and Matrix Exponent .

When the power Matrix Power and Matrix Exponent 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 Matrix Power and Matrix Exponent has an inverse matrix .

Example:
From matrix Matrix Power and Matrix Exponent we can obtain diagonal matrix Matrix Power and Matrix Exponent . Through modal matrix (augmented eigenvectors) Matrix Power and Matrix Exponent . The inverse modal matrix is Matrix Power and Matrix Exponent .

Then, matrix power can be computed through repeated matrix multiplication 4 times Matrix Power and Matrix Exponent . The same result can be obtained using diagonal matrix Matrix Power and Matrix Exponent . Then we multiply with modal matrix and its inverse, require only three matrix multiplications Matrix Power and Matrix Exponent .
Clearly this procedure is only more efficient than matrix multiplication when the power Matrix Power and Matrix Exponent is large.

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

Example:
Using the matrices in previous example, find Matrix Power and Matrix Exponent
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 Matrix Power and Matrix Exponent . We multiply back to obtain the matrix power Matrix Power and Matrix Exponent .


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

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