Schur Decomposition
From the book
Chapter 9: Matrix Decompositions and Beyond. In the chapter mind map this icon labels Schur Decomp: $A = QUQ^\top$. The discussion below is excerpted and lightly edited from § Theorem: Schur Decomposition in Mathematics for AI and Machine Learning.
For any $A \in \mathbb{C}^{N \times N}$, there exist unitary $Q$ and upper triangular $U$ such that $A = Q U Q^H$, with diagonal entries of $U$ being the eigenvalues of $A$. If $A$ has real eigenvalues, $Q$ and $U$ are real:
For real matrices with complex eigenvalues, the real Schur form has a quasi-upper-triangular $U$ with $2 \times 2$ blocks on the diagonal for complex conjugate pairs.
What this drawing shows
What you see. Represents triangularization by a unitary or orthogonal change of basis.
In the mind map. Chapter 9 — Schur Decomp. See From the book above for definitions, figures, and worked examples.
Where to read next
Read the full definitions, figures, and worked examples in Chapter 9: Matrix Decompositions and Beyond — see the mind-map node Schur Decomp.
