HW7 - Math 104 Fall 2009 Homework 7 Due Wednesday December...

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Math 104 Fall 2009 Homework 7 Due Wednesday, December 2, 2009 Do five of the following exercises. 1. If A is an n × n matrix with characteristic polynomial P ( λ ) = ( λ - λ 1 ) d 1 . . . ( λ - λ k ) d k what is the trace of A ? what is the determinant of A ? 2. A skew Hermitian matrix is a matrix obeying A * = - A . (a) Show that A = U Λ U * , with Λ diagonal and for a unitary U . (Hint: this is not a difficult question and you should think about how you could get back to the case you know; that is, the case where the matrix is Hermitian.) (b) Show that the eigenvalues are imaginary and the eigenvectors orthogonal. (c) Show that A + I is invertible. (d) Show that ( I - A )( I + A ) - 1 is an orthogonal matrix. 3. Suppose A is positive semidefinite. Can you a find a square root of this matrix? In other words, can you find a matrix B such that B 2 = A ? If yes, explain how you would construct it. If no, explain why no such matrix exists. 4. Suppose you have n vectors x 1 , . . . , x n in R m . In class, we have seen that the first principal component is the unit-normed vector u R m so that the projections of those vectors onto
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