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