Schur Decomposition

Schur Decomposition

Schur Decomposition (or “Schur Triangulation”) is a matrix decomposition technique:

• it decomposes a matrix $A = QTQ^T$ where
• $Q$ is orthogonal, i.e. $Q^T Q = I$
• $T$ is upper diagonal
• $T$ is called the Schur Form of $A$: it is upper triangular and has eivenvalues on the main diagonal
• $Q$’s columns contain the eigenvectors of $A$

Schur Form $T$

• we can write it down as $T = D + N$ where
• $D$ is diagonal with eigenvalues on the diagonal and
• $T$ is strictuly upper triangular

Symmetric Matrices

• if $A$ is symmetric, then $T$ is zero and there’s only the $D$ component
• then we have the Eigendecomposition

Computing Schur Decomposition:

• QR Algorithm is an iterative algorithm that does that