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 A_{1,1} to A_{n,n} for an n\times n 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 D of a matrix is equal to P^{-1}AP such that P is the matrix of eigenvectors (P=[v_1,\ldots,v_n]). Diagonal matrices are very useful, as computing determinants, products and sums of matrices, and powers becomes much simpler. For example, given the matrix A , A^n=PD^nP^{-1} .

Computation of the diagonal matrix

Given AP=PD , D can be found be making a diagonal matrix of the eigenvalues of A . P 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

\begin{vmatrix}\lambda I-\begin{bmatrix}1&2\\-1&4\end{bmatrix}\end{vmatrix}

Therefore D will be equal to


P 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 A .


Ad blocker interference detected!

Wikia is a free-to-use site that makes money from advertising. We have a modified experience for viewers using ad blockers

Wikia is not accessible if you’ve made further modifications. Remove the custom ad blocker rule(s) and the page will load as expected.