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Hints to Selected Problems in HW 5

二月 17, 2009

For 3. Answer the following mini questions:
(a) Let A = B + I, what is rank(A)?
(b) What is \chi_{A}(t)?
(c) If \lambda \in \sigma(A), what should be in \sigma(B) = \sigma(A - I)?

For 4, let v be an eigenvector of A with eigenvalue \lambda.
(a’)What is Av?
(a”)What is A^2v?
(a”’)Why can you deduce \lambda^2 - \lambda = 0?
(b’) Note that A satisfies the polynomial f(t) = t^2 - t. What can you say about the minimal polynomial m_A(t)?
(b”) Does the minimal polynomial have repeated roots?
(c’) We know from part (b) that A is diagonalizable, meaning that A is similar to a diagonal matrix D. What does the diagonal matrix look like?
(c”) Show that rank(D) = tr(D).
(c”’) Does similarity preserve rank and trace?

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