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. This transform was used in linear algebra so that it allow performing simpler calculations.

Matrix Diagonalization - dCode

Tag(s) : Matrix

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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. This transform was used in linear algebra so that it allow performing simpler calculations.

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

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

To diagonalize a matrix, a 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.

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 \( \begin{pmatrix} 1 \\ 0 \end{pmatrix} \) so it is not diagonalizable.

Calculate the inverse of the matrix \( P \)

Check that \( PDP^{-1} = M \)

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