Diagonalization

Similar matrices

2 square matrices and of the same order, are similar iff there exists an invertible matrix such that:

Similarity of 2 matrices is commutative.

Similar matrices have the set of eigenvalues.

Note

If and are similar, then and are similar.

Definition

A matrix is diagonalizable if it is similar to a diagonal matrix.

Here:

• is a diagonal matrix
• is an invertible matrix

Steps

• Find eigenvalues of :
• Find corresponding eigenvectors:
• Construct by joining the eigenvectors as columns

Note

Order of those eigenvectors is not a problem. Here the matrix differs based on the order, and hence is not unique.

Note

If is a real symmetric matrix with distinct eigenvalues then it has mutually orthogonal linearly independent eigenvectors. Hence the diagonalizing matrix (formed by using the normalized eigenvectors) is an orthogonal matrix.

Uses

Finding integer powers

Suppose is diagonalizable, and .