HW7_Sol - Math 104 Fall 2009 Homework 7 solution set We use...

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Math 104 Fall 2009 Homework 7: solution set We use the notations below to ease readability. Matrices are bold capital, vectors are bold lowercase and scalars or entries are not bold. For instance, A is a matrix and a ij its ( i, j )th entry. Likewise x is a vector and x j its j th component. The linear span generated by a group of vecotors v 1 , v 2 , . . . , v n is denoted by span( v 1 , v 2 , ..., v n ). Problem 1 Since the characteristic polynomial is P ( λ ) = ( λ - λ 1 ) d 1 · · · ( λ - λ k ) d k , we know all the eigenvalues are λ 1 , · · · , λ k with corresponding multiplicity d 1 , · · · , d k . Therefore, since the trace is the sum of the eigenvalues, trace( A ) = k i =1 d i λ i , and since the determinant is the product of the eigenvalues, det( A ) = Q k i =1 λ d i i . Problem 2 (a) Set B = i A . Since A * = - A , B * = - i A * = i A = B , which implies that B is Hermitian. By the spectral theorem, there is a unitary matrix U and a real diagonal matrix Λ such that B = U Λ U * , which implies that A = U ( i Λ ) U * . Therefore A is diagonalizable.
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