hw12 - Z n(g Find a quadratic polynomial f t(with...

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Math 110—Linear Algebra Fall 2009, Haiman Problem Set 12 Due Monday, Nov. 23, at the beginning of lecture. 1. Let J n denote the n × n matrix over R whose entries are all equal to 1. (a) Show that (1 , 1 ,..., 1) t is an eigenvector of J n . What is its eigenvalue? (b) Find the dimension of the nullspace of J n . (c) Use (a) and (b) to show that J n is diagonalizable, and find the diagonal matrix similar to J n . (d) Find the characteristic polynomial of J n . (e) Let Z n = J n - I n be the n × n matrix with zeroes on the diagonal and ones in all off-diagonal entries. Find det( Z n ), and show that Z n is invertible for n > 1. (f) Find the characteristic polynomial of
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Unformatted text preview: Z n . (g) Find a quadratic polynomial f ( t ) (with coefficients depending on n ) such that f ( Z n ) = 0. (h) Use (g) to calculate the inverse of Z n , expressed as a linear combination of Z n and I n . (This generalizes Problem Set 7, Problem 3.) 2. Let T : V → V be a linear operator, where V is finite dimensional. Suppose that W 1 ,...,W k are T-invariant subspaces of V such that T W i is diagonalizable for each i . Prove that if W 1 + ··· + W k = V , then T is diagonalizable. 3. Section 5.4, Exercises 13 and 20....
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