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lecture36

# lecture36 - Exercises December 3 2010 1 If A is an...

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Exercises December 3, 2010 1. If A is an invertible matrix and λ is an eigenvalue of A , then show that λ - 1 is an eigenvalue of A - 1 . Note that you must prove that λ 6 = 0 first. 2. What is the relationship between the eigenvalues of A and those of A T . 3. Let T : C n C n be a linear transformation, let λ be an eigenvalue of T , and let p be a polynomial. (a) Show that p ( λ ) is an eigenvalue of p ( T ). (b) Now suppose that μ is an eigenvalue of p ( T ), show that there is an eigenvalue λ of T such that p ( λ ) = μ . (c) Are these statements true for linear transformations on R n . 4. This exercise is about the generalization of the dot product to C n . To distinguish it from the usual dot product we will call it the inner product. If ~ z , ~w are vectors in C n , then we define the inner product by h ~ z, ~w i = z 1 w 1 + · · · + z n w n . Remmeber that a + ib = a - ib . For ~v, ~w, ~ z C n and α C , prove the following facts (a) h ~ z + ~w,~v i = h ~ z,~v i + h ~w,~v i (b) h α~ z, ~w i = α h ~ z, ~w i (c) h ~w, ~ z i = h ~v, ~ z i (d) h ~w, ~w i = 0 if and only if ~w = ~ 0.
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