Diagonalization

Similar Matrices

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

Properties

• Similarity of 2 matrices is commutative.
• Similar matrices have the set of eigenvalues.
• If $A$ and $B$ are similar, then $A^2$ and $B^2$ are similar.

Definition

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

Here:

• $D$ is a diagonal matrix
• $P$ is an invertible matrix

Steps

• Find eigenvalues of $A_{n\times n}$
  Say $\lambda_1,\lambda_2,\dots,\lambda_n$.
• Find corresponding eigenvectors
  Say $X_1,X_2,\dots,X_n$.
• Construct $P$ by joining the eigenvectors as columns

The matrix $P$ differs based on the order of the eigenvectors, and hence is not unique.

If $A$ is diagonalizable, its powers can be easily calculated. For $k\in \mathbb{R}$