Schur decomposition, proof of

The columns of the unitary matrixMathworldPlanetmath Q in Schur’s decomposition theorem form an orthonormal basisMathworldPlanetmath of n. The matrix A takes the upper-triangular form D+N on this basis. Conversely, if v1,,vn is an orthonormal basis for which A is of this form then the matrix Q with vi as its i-th column satisfies the theorem.

To find such a basis we proceed by inductionMathworldPlanetmath on n. For n=1 we can simply take Q=1. If n>1 then let vn be an eigenvectorMathworldPlanetmathPlanetmathPlanetmath of A of unit length and let V=v be its orthogonal complementMathworldPlanetmath. If π denotes the orthogonal projection onto the line spanned by v then (1-π)A maps V into V.

By induction there is an orthonormal basis v2,,vn of V for which (1-π)A takes the desired form on V. Now A=πA+(1-π)A so Avi(1-π)Avi(modv) for i{2,,n}. Then v,v2,,vn can be used as a basis for the Schur decompositionMathworldPlanetmath on n.

Title Schur decomposition, proof of
Canonical name SchurDecompositionProofOf
Date of creation 2013-03-22 14:04:01
Last modified on 2013-03-22 14:04:01
Owner mps (409)
Last modified by mps (409)
Numerical id 6
Author mps (409)
Entry type Proof
Classification msc 15-00