## Powers of Matrices

Given a matrixwith non zero entries only on the leading diagonal, it is very easy to find powers ofIf for example,then

A matrixis said to be diagonalizable if we can find a matrixsuch thator equivalentlywhereis a diagonal matrix with non zero entries only on the leading diagonal. Of course only square matrices can be diagonalized, but a wide range of square matrices can be diagonalized. If a matrix is diagonalizable, then

since when the brackets are removed each occurrence ofreduces to the identity matrix.

The matrixexists in the case when thetimesmatrixhasindependent eigenvectors. The eigenvectors form a linearly independent set, so a matrixwhose columns consist of them can be inverted thenfound using

Example: The matrixhas eigenvalues 1 and 3 with eigenvectorsandso thatand

D is the matrix with diagonal entries equal to the eigenvalues, with the eigenvalues in the column of that corresponding eigenvector:

is then given by