Schur decomposition
Undergraduate (real matrices with real eigenvalues)
Consider a square real matrix $A$ whose eigenvalues are all real. A Schur decomposition of $A$ is a factorization $A = Q T Q^{\top}$, where $Q$ is orthogonal and $T$ is upper triangular.
Graduate (complex matrices)
Consider a square complex matrix $A$. A Schur decomposition of $A$ is a factorization $A = Q T Q^$, where $Q$ is unitary ($Q^ Q = I$) and $T$ is upper triangular.
Examples
Worked examples
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The matrix \(\begin{bmatrix} 2 & 1 \\ 4 & 3 \end{bmatrix}\)
: \[T = \begin{bmatrix} \frac{5 + \sqrt{17}}{2} & -3 \\ 0 & \frac{5 - \sqrt{17}}{2} \end{bmatrix}\]
Undergraduate
In the Schur decomposition of \(A = \begin{bmatrix} 2 & 1 \\ 4 & 3 \end{bmatrix}\), the first column of $Q$ is a unit eigenvector for $\frac{5 + \sqrt{17}}{2}$, and the second column is perpendicular to it. The entry $-3$ above the diagonal is nonzero because $A$ is not normal.
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The matrix \(\begin{bmatrix} 4 & 2 \\ 2 & 3 \end{bmatrix}\)
: $T = \Lambda$, as in the eigendecomposition
Undergraduate
The matrix \(A = \begin{bmatrix} 4 & 2 \\ 2 & 3 \end{bmatrix}\) is symmetric positive definite, so it has every decomposition in the table, including a Cholesky decomposition. Its eigendecomposition, Schur decomposition, Jordan normal form, and singular value decomposition all coincide, and its polar and Toeplitz decompositions are trivial.
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The matrix \(\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}\)
: $Q = I$, $T = A$
Undergraduate
$A$ is upper triangular, so its LU decomposition is $I \cdot A$, and its QR decomposition is $I \cdot A$. $A$ is not diagonalizable, so its Jordan–Chevalley decomposition has a nonzero nilpotent part $N$.
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The matrix \(A = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}\)
: $T = \Lambda$, as in the eigendecomposition
Undergraduate
$A$ has no LU decomposition without exchanging rows, because its top-left entry is $0$, even though $A$ is invertible, but it has the PLU decomposition $A = A \cdot I \cdot I$. $A$ is orthogonal, so its QR decomposition is $A \cdot I$, its singular value decomposition is $A \cdot I \cdot I$, and its Jordan–Chevalley decomposition has $N = 0$ because $A$ is diagonalizable.
Properties
- Every square complex matrix has a Schur decomposition, but it is not unique. The diagonal entries of $T$ are the eigenvalues of $A$.
- The upper triangular factor $T$ is diagonal exactly when $A$ is normal ($A A^* = A^* A$). For a real symmetric matrix, a Schur decomposition is an eigendecomposition $A = Q \Lambda Q^{\top}$.
- Unlike an eigendecomposition, a Schur decomposition exists for every matrix and uses an orthogonal (or unitary) $Q$, so it can be computed stably. The QR algorithm for eigenvalues computes a Schur decomposition.
Notes
A real matrix with complex eigenvalues has a real Schur decomposition $A = Q T Q^{\top}$ with $Q$ real orthogonal, in which $T$ is upper triangular except for $2 \times 2$ blocks on its diagonal, one for each pair of complex-conjugate eigenvalues.
Distinctions involving this term
- Types of matrix decompositions: A matrix decomposition writes a matrix in terms of simpler matrices. Most are products: LU, PLU, and Cholesky decompositions use triangular factors, QR and singular value decompositions use orthogonal factors, and eigendecompositions, Schur decompositions, and Jordan normal forms are similarities $A = P M P^{-1}$ to a simpler matrix $M$ (diagonal, triangular, or made of Jordan blocks). In contrast, the Jordan–Chevalley decomposition is a sum of commuting diagonalizable and nilpotent parts, and the Toeplitz decomposition is a sum $H + K$ of a symmetric matrix $H$ and a skew-symmetric matrix $K$.