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extrahw-11

# extrahw-11 - x 1 and x 2 are the columns of S what are the...

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MATH 108 B FALL 2011 EXTRA-CREDIT PROBLEMS SHOW YOUR WORK CLEARLY. –1) Let A be a n × n skew-Hermitian matrix ( A * = A T = - A ). Prove: (a) The eigenvalues of A are pure imaginary, and eigenvectors corresponding to different eigenvalues are orthogonal. (we recall that if x, y C n , then h x, y i = x T y = n j =1 x j y j ) (b) Prove that ( I - A ) and ( I + A ) are invertible. Typeset by A M S -T E X 1

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2 MATH 108 B FALL 2011 EXTRA-CREDIT PROBLEMS (c) Prove that e A defined below is an unitary matrix (i.e. U unitary if U * U = I ). e A = X n =0 A n n ! . (d) Prove that Q = ( I - A )( I + A ) - 1 is an unitary matrix. (e) If A = 0 2 - 2 0 compute Q as in part (d).
MATH 108 B FALL 2011 EXTRA-CREDIT PROBLEMS 3 –2) If the vectors

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Unformatted text preview: x 1 and x 2 are the columns of S , what are the eigenvalues and eigenvectors of B = S ± 2 1 ² S-1 , C = S ± 2 3 1 ² S-1 ? 4 MATH 108 B FALL 2011 EXTRA-CREDIT PROBLEMS –3) (a) Show that if B is unitary and λ ∈ C is an eigenvalue of B , then | λ | = 1. (b) Show that if A is normal (i.e. A * A = AA * ) and invertible, then B = A * A-1 is unitary. MATH 108 B FALL 2011 EXTRA-CREDIT PROBLEMS 5 –4) Show that A = ± 1 ² . has no square root, i.e. there is no B such that B 2 = A ....
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