proof that commuting matrices are simultaneously triangularizable
Proof by induction on , order of matrix.
For we can simply take . We assume that there exists a common unitary matrix that triangularizes simultaneously commuting matrices ,.
So we have to show that the statement is valid for commuting matrices, . From hypothesis and are commuting matrices so these matrices have a common eigenvector.
Let , where be the common eigenvector of unit length and , are the eigenvalues of and respectively. Consider the matrix, where be orthogonal complement of and , then we have that
It is obvious that the above matrices and also , , matrices are commuting matrices. Let and then there exists unitary matrix such that Now is a unitary matrix, and we have
Analogously we have that
|Title||proof that commuting matrices are simultaneously triangularizable|
|Date of creation||2013-03-22 15:27:08|
|Last modified on||2013-03-22 15:27:08|
|Last modified by||georgiosl (7242)|