Schur decomposition, proof of
The columns of the unitary matrix in Schur’s decomposition theorem form an orthonormal basis of . The matrix takes the upper-triangular form on this basis. Conversely, if is an orthonormal basis for which is of this form then the matrix with as its -th column satisfies the theorem.
To find such a basis we proceed by induction on . For we can simply take . If then let be an eigenvector of of unit length and let be its orthogonal complement. If denotes the orthogonal projection onto the line spanned by then maps into .
By induction there is an orthonormal basis of for which takes the desired form on . Now so for . Then can be used as a basis for the Schur decomposition on .
|Title||Schur decomposition, proof of|
|Date of creation||2013-03-22 14:04:01|
|Last modified on||2013-03-22 14:04:01|
|Last modified by||mps (409)|