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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) -

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# Schur Decomposition (Matrix)

## Schur Decomposition Calculator

 Result Format Automatic Selection Exact Value (when possible) Approximate Numerical Value Scientific Notation

### 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). The matrix T is a trigonalization (or triangulation).

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

There is always a decomposition of Schur, all matrices are trigonalizable, but not uniquely.

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

### Why using the Schur Decomposition?

The Schur decomposition makes it possible to simplify the form of the matrices and thus to facilitate the resolution of linear equations or any other problem using the matrix.

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