proof that commuting matrices are simultaneously triangularizable

Proof by induction on n, order of matrix.
For n=1 we can simply take Q=1. We assume that there exists a common unitary matrixMathworldPlanetmath S that triangularizes simultaneously commuting matricesMathworldPlanetmath ,(n-1)×(n-1).
So we have to show that the statement is valid for commuting matrices, n×n. From hypothesisMathworldPlanetmathPlanetmath A and B are commuting matrices n×n so these matrices have a common eigenvectorMathworldPlanetmathPlanetmathPlanetmath.
Let Ax=λx, Bx=μx where x be the common eigenvector of unit length and λ, μ are the eigenvaluesMathworldPlanetmathPlanetmathPlanetmathPlanetmath of A and B respectively. Consider the matrix, R=(xX) where X be orthogonal complementMathworldPlanetmathPlanetmath of x and RHR=I, then we have that


It is obvious that the above matrices and also XHBX, XHAX ,(n-1)×(n-1) matrices are commuting matrices. Let B1=XHBX and A1=XHAX then there exists unitary matrix S such that SHB1S=T¯2,SHA1S=T¯1. Now Q=R(100S) is a unitary matrix, QHQ=I and we have


Analogously we have that

Title proof that commuting matrices are simultaneously triangularizable
Canonical name ProofThatCommutingMatricesAreSimultaneouslyTriangularizable
Date of creation 2013-03-22 15:27:08
Last modified on 2013-03-22 15:27:08
Owner georgiosl (7242)
Last modified by georgiosl (7242)
Numerical id 10
Author georgiosl (7242)
Entry type Proof
Classification msc 15A23