Diagonalization is the process of finding a corresponding diagonal matrix (a matrix in which the only non-zero components are on the diagonal line from to for an matrix) for a given diagonalizable matrix. A matrix is diagonalizable if and only if the matrix of eigenvectors is invertable (that is, the determinant does not equal zero). If a matrix is not diagonalizable, is is called a defective matrix.
The diagonal of a matrix is equal to such that is the matrix of eigenvectors (). Diagonal matrices are very useful, as computing determinants, products and sums of matrices, and powers becomes much simpler. For example, given the matrix , .
Computation of the diagonal matrix
Given , can be found be making a diagonal matrix of the eigenvalues of . will be equal to the matrix of corresponding eigenvectors. For example, say we have the matrix
To find the eigenvalues, we must first find the characteristic polynomial, which will be equal to
Therefore will be equal to
will be the matrix of eigenvectors corresponding to the above diagonal matrix. The eigenvectors will be the non-trivial solution to
This is useful to us because, among other things, we can use this to find large powers of .