Give the best answer to the following questions using

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3 (8 points) Give the best answer to the following questions, using methods from Chapter 5. Include a brief justification. (a) If we know an eigenvector v R n of A R n × n sym , how can we find the corre- sponding eigenvalue? (b) If A R n × n sym is tridiagonal, how can we find the eigenvalues without solving the characteristic polynomial? (c) If A ( k ) is produced by Jacobi’s method, what is happening to the entries of A ( k ) as k and what does this tell us about the eigenvalues of A ?
MATH???? 4 (9 points) (a) (3 points) Write down Gerschgorin’s first theorem. (b) (3 points) Write down a 4 × 4 matrix, without any zero entries, that has four distinct eigenvalues, all in the first quadrant of C . (c) (3 points) Let A ( k ) be the sequence of matrices produced by Jacobi’s eigen- value method with A ( 0 ) = A . Recall that lim k n i , j = 1 , i 6 = j | A ( k ) i j | 2 = 0 . Using Gerschgorin’s theorems, explain why the diagonal elements of A ( k ) converge to the eigenvalues of A .
MATH???? 5 (12 points) (a) (3 points) What is the Householder matrix corresponding to v = ( 1 , 0 , - 1 , 2 ) T ? (b) (3 points) Let H be the Householder matrix corresponding to v R n , show that H is orthogonal. (c) (6 points) Use Householder’s method to transform A = 4 3 - 4 3 2 0 - 4 0 1 into a tridiagonal matrix.
MATH???? 6 (9 points)
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