Tool to calculate Schur decomposition (or Schur triangulation) that makes it possible to write any numerical square matrix into a multiplication of a unitary matrix and an upper triangular matrix.

Schur Decomposition (Matrix) - dCode

Tag(s) : Matrix

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

A suggestion ? a feedback ? a bug ? an idea ? *Write to dCode*!

Sponsored ads

Tool to calculate Schur decomposition (or Schur triangulation) that makes it possible to write any numerical square matrix into a multiplication of a unitary matrix and an upper triangular matrix.

The **Schur decomposition** of a square matrix $ M $ is its writing in the following form (also called Schur form): $$ M = Q.T.Q^{-1} $$

with $ Q $ a unitary matrix (such as $ Q^*.Q=I $) and $ T $ is an upper triangular matrix whose diagonal values are the eigenvalues of the matrix.

This decomposition only applies to numerical square matrices (no variables)

Example: The Schur triangulation of the matrix $ M = \begin{pmatrix} 1 & 3 \\ 2 & 4 \end{pmatrix} $ gives $$ Q = \begin{pmatrix} −0.825 & 0.566 \\ 0.566 & −0.825 \end{pmatrix}, T = \begin{pmatrix} −0.372 & −1 \\ 0 & 5.372 \end{pmatrix} $$

There is always a decomposition of Schur, but it is not necessarily unique.

dCode uses computer algorithms involving QR decomposition.

Manually, find a proper vector $ u_1 $ of the matrix $ M $ by calculating its eigenvalues $ \Lambda_i $. Calculate its normalized value and an orthonormal basis $ {u_1, v_2} $ to obtain $ U = [u_1, v_2] $. Express the matrix $ M $ in the orthonormal basis $ A_{{u_1, v_2}} = U^{-1}.A.U = U^{T}.A.U $. Repeat the operation for each eigenvector to obtain the triangular matrix. NB: for a 2x2 matrix, only one operation is necessary and $ T = A_{{u_1, v_2}} $

dCode retains ownership of the source code of the script Schur Decomposition (Matrix) 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 released for free. To download the online Schur Decomposition (Matrix) script for offline use on PC, iPhone or Android, ask for price quote on contact page !

schur,matrix,triangulation,decomposition,unitary,triangular

Source : https://www.dcode.fr/matrix-schur-decomposition

© 2019 dCode — The ultimate 'toolkit' to solve every games / riddles / geocaches. dCode

Feedback

▲