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.

Matrix Diagonalization - dCode

Tag(s) : Matrix

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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.

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

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

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.

Calculate the inverse of the matrix $ P $

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

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