Matrix Diagonalization

Question 1

What is a diagonal matrix? (Definition)

Answer

A diagonal matrix is a matrix whose elements out of the trace (the main diagonal) are all null (zeros).

A square matrix $M$ is diagonal if $M_{i,j} = 0$ for all $i \neq j$ .

Example: A diagonal matrix: $\begin{bmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{bmatrix}$

Diagonalization is a transform used in linear algebra usually to simplify calculations (like powers of matrices).

Question 2

What is a diagonalizable matrix? (Definition)

Answer

A matrix is diagonalizable if there exists an invertible matrix $P$ and a diagonal matrix $D$ such that $M = PDP^{-1}$

Question 3

How to diagonalize a matrix?

Answer

To diagonalize a matrix, a diagonalisation method consists in calculating its eigenvectors and its eigenvalues.

Example: The matrix $M = \begin{bmatrix} 1 & 2 \\ 2 & 1 \end{bmatrix}$ has for eigenvalues $3$ and $-1$ and eigenvectors respectively $\begin{bmatrix} 1 \\ 1 \end{bmatrix}$ and $\begin{bmatrix} -1 \\ 1 \end{bmatrix}$

The diagonal matrix $D$ is composed of eigenvalues.

Example: $D = \begin{bmatrix} 3 & 0 \\ 0 & -1 \end{bmatrix}$

The order of the eigenvalues in $D$ has no intrinsic importance, but it must match the order of the eigenvectors in $P$ .

The invertible matrix $P$ is composed of the eigenvectors respectively in the same order of the columns than its associated eigenvalues.

P must be a normalized matrix.

Example: $P = \begin{bmatrix} 1 & -1 \\ 1 & 1 \end{bmatrix}$ Normalization of P: $P = \frac{1}{\sqrt{2}} \begin{bmatrix} 1 & -1 \\ 1 & 1 \end{bmatrix}$

Question 4

How to prove that a matrix is not diagonalizable?

Answer

A matrix is not diagonalizable if it does not have as many dimensions as distinct eigenvectors.

Example: The matrix of dimension 2: $M = \begin{bmatrix} 5 & 1 \\ 0 & 5 \end{bmatrix}$ has a double eigenvalue: $5$ and therefore a single eigenvector $\begin{bmatrix} 1 \\ 0 \end{bmatrix}$ so it is not diagonalizable.

Example: A 3x3 matrix with a triple eigenvalue therefore a single eigenvector is not diagonalizable.

Question 5

How to check a diagonalized matrix calculation?

Answer

Calculate the inverse of the matrix $P$

Diagonalization should give $PDP^{-1} = M$

Question 6

What is orthogonal diagonalization?

Answer

Orthogonal diagonalization is a specific type of matrix diagonalization applicable to symmetric matrices. It consists of expressing a symmetric matrix $M$ as $M = Q D Q ^ T$ or the product of an orthogonal matrix $Q$ (so $Q.Q^T = Q^T.Q = I$ ), a diagonal matrix $D$ , and the transpose of $Q$ , denoted $Q^T$ .

Orthogonal diagonalization is useful because it preserves the orthogonality of the eigenvectors and can simplify computations involving symmetric matrices.

Matrix Diagonalization