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

dCode is free and its tools are a valuable help in games, puzzles and problems to solve every day!

You have a problem, an idea for a project, a specific need and dCode can not (yet) help you? You need custom development? *Contact-me*!

Sponsored ads

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

dCode retains ownership of the source code of the script Matrix Diagonalization online. Except explicit open source licence (indicated Creative Commons / free), any algorithm, applet, snippet, software (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, translator), or any function (convert, solve, decrypt, encrypt, decipher, cipher, decode, code, translate) written in any informatic langauge (PHP, Java, C#, Python, Javascript, Matlab, etc.) which dCode owns rights will not be given for free. So if you need to download the online Matrix Diagonalization script for offline use, check contact page !

diagonalization,diagonal,matrix,endomorphism,diagonalized,diagonalizable

Source : https://www.dcode.fr/matrix-diagonalization

© 2018 dCode — The ultimate 'toolkit' to solve every games / riddles / geocaches. dCode