Definition of Schur triangulation

Definition: Schur triangulation (or Schur decomposition) states that every square matrix $A$ over $\mathbb{C}$ can be written as:

$$A = QTQ^*$$

where $Q$ is unitary and $T$ is upper triangular. The diagonal entries of $T$ are the eigenvalues of $A$.

For real matrices with real eigenvalues: $A = Q T Q^T$ where $Q$ is orthogonal and $T$ is upper triangular.

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.

Applications:

• Computing matrix functions: $f(A) = Q f(T) Q^*$
• Theoretical foundation for many eigenvalue algorithms
• Proving properties about eigenvalues and eigenvectors

The Schur decomposition always exists (over $\mathbb{C}$) and is computed numerically via the QR algorithm.