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.

Real symmetric matrix

Suppose is a real symmetric matrix . If it has 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 .