proof of Wielandt-Hoffman theorem

Since both A and B are normal, they can be diagonalized by unitary transformationsMathworldPlanetmath:


where C and D are diagonalMathworldPlanetmath, V and W are unitary, and () denotes the conjugate transposeMathworldPlanetmath. The Frobenius matrix norm is defined by the quadratic form AF2=tr[AA] and is invariant under unitary transformations, hence


where U=WV. The matrix U is also unitary, let its matrix elements be given by (U)ij=uij. Unitarity implies that the matrix with elements |uij|2 has its row and column sums equal to 1, in other words, it is doubly stochastic.

The diagonal elements Cii=ai are eigenvaluesMathworldPlanetmathPlanetmathPlanetmathPlanetmath of A and Dii=bi are those of B. Writing out the Frobenius normMathworldPlanetmath explicitly, we get


where the minimum is taken over all doubly stochastic matrices S, whose elements are (S)ij=sij. By the Birkoff-von Neumann theorem, doubly stochastic matrices form a closed convex ( polyhedronMathworldPlanetmath with permutation matricesMathworldPlanetmath at the vertices. The expression ija¯isijbj is a linear functionalMathworldPlanetmath on this polyhedron, hence its minimum is achieved at one of the vertices, that is when S is a permutation matrix.

If S represents the permutation σ, its action can be written as jsijbj=bσ(i). Finally, we can write the last inequalityMathworldPlanetmath as


which is exactly the statement of the Wielandt-Hoffman theorem.

Title proof of Wielandt-Hoffman theorem
Canonical name ProofOfWielandtHoffmanTheorem
Date of creation 2013-03-22 14:58:51
Last modified on 2013-03-22 14:58:51
Owner Andrea Ambrosio (7332)
Last modified by Andrea Ambrosio (7332)
Numerical id 6
Author Andrea Ambrosio (7332)
Entry type Proof
Classification msc 15A18
Classification msc 15A42