proof of Wielandt-Hoffman theorem
Since both and are normal, they can be diagonalized by unitary transformations:
where . The matrix is also unitary, let its matrix elements be given by . Unitarity implies that the matrix with elements has its row and column sums equal to , in other words, it is doubly stochastic.
where the minimum is taken over all doubly stochastic matrices , whose elements are . By the Birkoff-von Neumann theorem, doubly stochastic matrices form a closed convex (http://planetmath.org/ConvexSet) polyhedron with permutation matrices at the vertices. The expression is a linear functional on this polyhedron, hence its minimum is achieved at one of the vertices, that is when is a permutation matrix.
which is exactly the statement of the Wielandt-Hoffman theorem.
|Title||proof of Wielandt-Hoffman theorem|
|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)|
|Author||Andrea Ambrosio (7332)|