## Matrix Diagonalization

## Answers to Questions

### What is a diagonal matrix? (Definition)

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

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

### What is a diagonalizable matrix? (Definition)

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

### How to diagonalize a matrix?

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{pmatrix} 1 \\ 1 \end{pmatrix} $ and $ \begin{pmatrix} -1 \\ 1 \end{pmatrix} $

The diagonal matrix $ D $ is composed of eigenvalues.

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

The invertable 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: $$ P = \frac{1}{\sqrt{2}} \begin{bmatrix} 1 & -1 \\ 1 & 1 \end{bmatrix} $$

### How to prove that a matrix is not diagonalizable?

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{pmatrix} 1 \\ 0 \end{pmatrix} $ so it is not diagonalizable.

### How to check a diagonalized matrix calculation?

Calculate the inverse of the matrix $ P $

Check that $ PDP^{-1} = M $

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