Unformatted text preview: 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 − λ1 )d1 . . . (λ − λk )dk 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 ΛU ∗ , 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 x1 , . . . , xn in Rm . In class, we have seen that the ﬁrst principal component is the unit-normed vector u ∈ Rm so that the projections of those vectors onto u have maximum variance. Another way to look at this is as follows: consider a line L going through some point x0 ∈ Rm and with some orientation u ∈ Rm , u = 1 (the equation of this line is x0 + tu where t is a scalar). Now consider the line that is closest to the point in the sense that it minimizes
n |distance(xi , L)|2
i=1 (the sum of squares of the distances between the xi ’s and the line). (a) Show that the slope of the closest line is the ﬁrst principal component. (b) Show that this line goes through the average vector x = ¯
n i=1 xi . 5. Problem 23.1 in Trefethen and Bau. (In this exercise U is the upper triangular factor so that with L = U ∗ , A∗ A = U ∗ U = LL∗ .) 6. Suppose A is positive semideﬁnite. Show that the maximum eigenvalue of A, denoted by λmax , is given by the so-called Rayleigh quotient w∗ Aw sup ∗ . w=0 w w 1 ...
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