211a6 - [1(4 Let A = 1-3 3 3-5 3 6-6 4 Find the eigenvalues...

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MATH 211 FALL 2010, ASSIGNMENT SIX (DUE FRIDAY, DEC. 3 IN CLASS BEFORE THE LECTURE BEGINS) Show your work neatly and clearly. Illegible or disorganized solutions will receive no credit. (1) Show that A and A T have the same eigenvalues. Are the eigenvalues of A and of A - 1 related at all? [3] (2) Let S be the subspace of R 3 spanned by vectors (1 , - 1 , 2) and (0 , 1 , 1) and let P be the 3 by 3 projection matrix onto the space S . Find all the eigenvalues for P and a basis for each corresponding eigenspace. This requires very little calculation! Does R 3 have an orthonormal basis consisting of eigenvectors of P ? Explain. [4] (3) Let A be a 3 by 3 matrix with eigenvalues - 3 , 1 and 2. Find each of the following (with justi±cations): a) N ( A ). [1] b) rank ( A - I ). [1] c) det A . [1] d) trace ( A - 1 ). [1] e) trace (( A - 2 I ) T ).
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Unformatted text preview: [1] (4) Let A = 1-3 3 3-5 3 6-6 4 . Find the eigenvalues of A and a basis for each of the associated eigenspaces. [4] (5) Diagonalize the following matrix. -2-1 2 3 2 [4] (6) Let the sequence a , a 1 , a 2 , ··· be given by a = 0 , a 1 = 1 , and a k = ( a k-1 + a k-2 ) / 2 for k ≥ 2 . Find the matrix that can be used to generate this sequence as we used a matrix to generate the Fibonacci sequence. Apply the diagonalization technique to ±nd an explicit formula for the sequence. [4] (7) Find an orthogonal matrix Q that diagonalizes the symmetric matrix 1 1 3 1 3 1 3 1 1 . [4] Goes up to the end of § 6.4. 1...
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