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PS2 - 1 2 equals trM and that 1 2 equals detM(the...

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MATH 580: Problem Set 2 due Thursday, September 23, 2004 1. Show that the following are or are not linear vector spaces, and provide a basis set for each if appropriate: a) the set of all symmetric 3 x 3 matrices; b) the set of all 4th order polynomials in x equal to zero at x = 0; c) the set of all 4th order polynomials in x equal to one at x = 0. 2. Using the definition of an inner product, show that (a) the induced norm is real. (b) h u, v + w i = h u, v i + h u, w i (c) the function h x, y i = Σ x i ¯ y i for n -tuples is a valid inner product. 3. Find the eigenvalues λ i and eigenvectors for the following matrices: M = ± 2 1 6 3 ² M = ± c b b c ² . For each confirm that
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Unformatted text preview: λ 1 + λ 2 equals trM , and that λ 1 λ 2 equals detM (the two invariants of M ). For the second matrix, what conditions on b and c will make M positive definite? 4. Keener 1.2.4 (p.51) 5. If A is a Hermitian matrix, show that 1) all eigenvalues are real, 2) if two eigenvalues are distinct, then the corresponding eigenvectors are orthogonal. 6. Show that A * = ¯ A T for h x, y i = Σ x i ¯ y i 7. Keener 1.2.5 (p.51) 8. If A and B are positive definite matrices and Ax = λBx , where λ is a scalar, show that λ must be positive....
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