assig5 - MATH 245 Linear Algebra 2, Assignment 5 Due Wed...

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Unformatted text preview: MATH 245 Linear Algebra 2, Assignment 5 Due Wed Nov 24 1: (a) Let A = 1 2 1 2 4 2 1 2 1 . Find an orthogonal matrix P and a diagonal matrix D such that P * AP = D . (b) Let A = 2 i- 2 + i 1- i 3 . Find a unitary matrix P and an upper-triangular matrix T so that P * AP = T . 2: (a) Let U be a (possibly infinite-dimensional) inner product space over R or C , and let L : U U be linear. Show that if L * = L then all of the eigenvalues of L are real, and any two eigenvectors associated to distinct eigenvalues are orthogonal. (b) Let U be a finite-dimensional inner product space over R or C , and let L : U U be linear. Show that the following are equivalent. 1. L * = L and h L ( x ) ,x i 0 for all x U . 2. L * = L and all of the eigenvalues of L are non-negative real numbers. 3. L = M * M for some linear map M : U U . 3: Let A M n n ( C ). Let 1 , 2 , , n be the eigenvalues of A (listed with repetition according to algebraic multiplicity). Show that the following are equivalent.multiplicity)....
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