SCHUR DECOMPOSITION

In the mathematical discipline of linear algebra, 'the Schur decomposition' or 'Schur triangulation' (named after Issai Schur) is an important matrix decomposition.

Contents

Statement

Proof

Notes

Applications

References

Statement

The Schur decomposition reads as follows: if ''A'' is a ''n'' × ''n'' square matrix with complex entries, then ''A'' can be expressed as
:$A\; =\; Q\; U\; Q^$
★ ,
where ''Q'' is a unitary matrix, ''Q
★ '' is the conjugate transpose of ''Q'', and ''U'' is an upper triangular matrix. It will be clear from the proof that the diagonal entries of U are exactly the eigenvalues of ''A''.
In other words, the Schur decomposition claims that there exist a nested sequence of ''A''-invariant subspaces {0} = ''V''₀ ⊂ ''V''₁ ⊂ ... ''V_m'' ⊂ 'C'^''n'', where ''m'' ≤ ''n''. Phrased somewhat differently, we say that an operator ''T'' on a complex finite-dimensional vector space stabilizes a
flag {''V''₁,...,''V_m''}.

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 spanning ''V_λ'' and ''V_λ''^⊥ respectively)
:
egin{matrix}
V_{lambda} \
oplus \
V_{lambda}^{perp}
end{matrix}
ightarrow
egin{matrix}
V_{lambda} \
oplus \
V_{lambda}^{perp}
end{matrix}

where ''λ I_λ'' is the identity operator on ''V_λ''. The above matrix would be upper-triangular except for the ''A''₂₂ block. But exactly the same procedure can be applied to the sub-matrix ''A''₂₂, viewed as an operator on ''V_λ''^⊥, and its submatrices. Continue this way until we exhaust the space 'C'^''n'' gives 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 'C'^''n'' modulo ''V_λ''. This operator is precisely the ''A''₂₂ submatrix from above. As before, ''T'' would have an eigenspace, say ''W_μ'' ⊂ 'C'^''n'' 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.

Notes

Although every square matrix has a Schur decomposition, in general this decomposition is not unique. For example, the eigenspace ''V_λ'' can have dimension > 1, in which case any orthonormal basis for ''V_λ'' would lead to the desired result.
Write the triangular matrix ''U'' as ''U'' = ''D'' + ''N'', where ''D'' is diagonal and ''N'' is strictly upper triangular (and thus nilpotent). The diagonal matrix ''D'' contains the eigenvalues of ''A'' in arbitrary order. The nilpotent part ''N'' is generally not unique either, but its Frobenius norm is uniquely determined by ''A''.
It is clear that if ''A'' is a normal matrix, then ''U'' from its Schur decomposition must be a diagonal matrix and the column vectors of ''Q'' are the eigenvectors of ''A''. Therefore, the Schur decomposition extends the spectral decomposition. In particular, if ''A'' is positive definite, the Schur decomposition of ''A'', its spectral decomposition, and its singular value decomposition coincide.
A commuting family {''A_i''} of matrices can be simultaneously triangularized, i.e. there exists a unitary matrix ''Q'' such that, for every ''A_i'' in the given family, ''Q A_i Q
★ '' is upper triangular. This can be readily deduced from the above proof. Take element ''A'' from {''A_i''} and again consider a eigenspace ''V_A''. Then ''V_A'' is invariant under all matrices in {''A_i''}. Therefore all all matrices in {''A_i''} must share one common eigenvector in ''V_A''. Induction then proves the claim. As a corollary, we have that every commuting family of normal matrices can be simultaneously diagonalized.
In the infinite dimensional setting, not every bounded operator on a Banach space has an invariant subspace. However, the upper-triangularization of an arbitrary square matrix does generalize to compact operators. Every compact operator on a complex Banach space has a nest of closed invariant subspaces.

Applications

Lie-theory applications include:

★ Every invertible operator is contained in a Borel group.

★ Every operator fixes a point of the flag manifold.

References

★ Roger A. Horn and Charles R. Johnson, ''Matrix Analysis,'' Sections 2.3 and further, Cambridge University Press, 1985. ISBN 0-521-38632-2.