# solutions4.2 - Math 307 Problems for section 4.2 1(i What...

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Math 307: Problems for section 4.2 November 14, 2012 1. (i) What can you say about the diagonal elements of a Hermition matrix? (ii) Show that if A is an n × n matrix such that ( v , A w ) = ( A v , w ) then A is Hermitian. (i) Diagonal entries of Hermitian matrices are real, because for a Hermitian matrix A = [ a i,j ], we have a i,i = a i,i . (ii) The condition can be written ( v , A w ) = ( v , A * w ) . Taking v = e i and w = e j we find that a i,j = ( e i , A e j ) = ( e i , A * e j ) = a i,j . 2. Show that if A is any matrix then A * A and AA * are Hermitian with non-negative eigen- values.