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Matrix Diagonalization

Tool to diagonalize a matrix. The diagonalization of a matrix consists of writing it in a base where its elements outside the diagonal are null.

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Matrix Diagonalization -

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# Matrix Diagonalization

## Matrix Diagonalization

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

Diagonalization is a transform used in linear algebra so that it allows performing easier calculations.

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

A matrix is diagonalizable if there exists an invertible 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 $\begin{bmatrix} 1 \\ 1 \end{bmatrix}$ 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 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: $$P = \frac{1}{\sqrt{2}} \begin{bmatrix} 1 & -1 \\ 1 & 1 \end{bmatrix}$$

### 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 $\begin{bmatrix} 1 \\ 0 \end{bmatrix}$ so it is not diagonalizable.

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

### How to check a diagonalized matrix calculation?

Calculate the inverse of the matrix $P$

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

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