Schur decomposition

In the mathematical discipline of linear algebra, the Schur decomposition or Schur triangulation, named after Issai Schur, is a matrix decomposition. It allows one to write an arbitrary complex square matrix as unitarily equivalent to an upper triangular matrix whose diagonal elements are the eigenvalues of the original matrix.

Statement

The Schur decomposition reads as follows: if A is an n × n square matrix with complex entries, then A can be expressed as

where Q is a unitary matrix (so that its inverse Q⁻¹ is also the conjugate transpose Q* of Q), and U is an upper triangular matrix, which is called a Schur form of A. Since U is similar to A, it has the same spectrum, and since it is triangular, its eigenvalues are the diagonal entries of U.

The Schur decomposition implies that there exists a nested sequence of A-invariant subspaces {0} = V₀ ⊂ V₁ ⊂ ⋯ ⊂ V_n = Cⁿ, and that there exists an ordered orthonormal basis (for the standard Hermitian form of Cⁿ) such that the first i basis vectors span V_i for each i occurring in the nested sequence. Phrased somewhat differently, the first part says that a linear operator J on a complex finite-dimensional vector space stabilizes a complete flag (V₁, ..., V_n).

Proof

A constructive proof for the Schur decomposition is as follows: every operator A on a complex finite-dimensional vector space has an eigenvalue λ, corresponding to some eigenspace V_λ. Let V_λ^⊥ be its orthogonal complement. It is clear that, with respect to this orthogonal decomposition, A has matrix representation (one can pick here any orthonormal bases Z₁ and Z₂ spanning V_λ and V_λ^⊥ respectively)

where I_λ is the identity operator on V_λ. The above matrix would be upper-triangular except for the A₂₂ block. But exactly the same procedure can be applied to the sub-matrix A₂₂, viewed as an operator on V_λ^⊥, and its submatrices. Continue this way until the resulting matrix is upper triangular. Since each conjugation increases the dimension of the upper-triangular block by at least one, this process takes at most n steps. Thus the space Cⁿ will be exhausted and the procedure has yielded the desired result.

The above argument can be slightly restated as follows: let λ be an eigenvalue of A, corresponding to some eigenspace V_λ. A induces an operator T on the quotient space Cⁿ/V_λ. This operator is precisely the A₂₂ submatrix from above. As before, T would have an eigenspace, say W_μ ⊂ Cⁿ modulo V_λ. Notice the preimage of W_μ under the quotient map is an invariant subspace of A that contains V_λ. Continue this way until the resulting quotient space has dimension 0. Then the successive preimages of the eigenspaces found at each step form a flag that A stabilizes.