Diagonalization

Similar matrices

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

$$B=P^{-1}AP$$

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. $$

$$\exists\, D,P\;\text{ s.t. } D = P^{-1}AP$$

Here:

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

Steps

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

$$P=(X_1 X_2 \dots X_n)_{n\times n} \;\; \land \;\; D= \begin{pmatrix} \lambda_1 \\ & \ddots \\ & & \lambda_n \\ \end{pmatrix}$$

Note

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

Real symmetric matrix

Suppose $A_{n\times n}$ is a real symmetric matrix . If it has distinct eigenvalues then it has $n$ mutually orthogonal linearly-independent eigenvectors.

Hence the diagonalizing matrix $P$ (formed by using the normalized eigenvectors) is an orthogonal matrix. $$

Uses

Finding integer powers

Suppose $A_{n\times n}$ is diagonalizable, and $k\in\mathbb{R}$.

$$A = P^{-1}DP \implies A^k = P^{-1}D^kP$$