# Definition of Schur triangulation

Put content here**Definition:** **Schur triangulation** (or **Schur decomposition**) states that every square matrix $A$ over $\mathbb{C}$ can be written as:
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$$A = QTQ^*$$
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where $Q$ is unitary and $T$ is upper triangular. The diagonal entries of $T$ are the eigenvalues of $A$.
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For real matrices with real eigenvalues: $A = Q T Q^T$ where $Q$ is orthogonal and $T$ is upper triangular.
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**Proof idea:** By induction on dimension. Pick an eigenvector $v$, extend to an orthonormal basis, and the matrix in this basis has the form with eigenvalue in the (1,1) position and a smaller block below.
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**Applications:**
- Computing matrix functions: $f(A) = Q f(T) Q^*$
- Theoretical foundation for many eigenvalue algorithms
- Proving properties about eigenvalues and eigenvectors
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The Schur decomposition always exists (over $\mathbb{C}$) and is computed numerically via the QR algorithm.

# Parents

* Factorization of matrices