Search for a tool
Schur Decomposition (Matrix)

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.

Results

Schur Decomposition (Matrix) -

Tag(s) : Matrix

Share dCode and you

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!

Please, check our community Discord for help requests!

Thanks to your feedback and relevant comments, dCode has developped the best Schur Decomposition (Matrix) tool, so feel free to write! Thank you !

# Schur Decomposition (Matrix)

## Schur Decomposition Calculator

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.

### What is the Schur Decomposition ? (Definition)

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.

### How to calculate the Schur Decomposition for a matrix?

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 $T = A_{{u_1, v_2}}$

## Source code

dCode retains ownership of the online 'Schur Decomposition (Matrix)' tool source code. Except explicit open source licence (indicated CC / Creative Commons / free), any algorithm, applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, translator), or any function (convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (PHP, Java, C#, Python, Javascript, Matlab, etc.) no data, script or API access will be for free, same for Schur Decomposition (Matrix) download for offline use on PC, tablet, iPhone or Android !

## Need Help ?

Please, check our community Discord for help requests!