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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.

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

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