Math 191 Sample Midterm 2B Spring 2023
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1a. (4 points) Let A = USVT be a full SVD of an arbitrary m×n real matrix A. Show that the singular values σi = Sii do not depend on which orthogonal matrices U and V are used.
1b. (4 points) Suppose A is an m × n matrix with real entries. Prove that ∥A∥2 = σ1, where σ1 is the largest singular value of A and we recall the definition
∥A∥2 = max ∥Ax∥2 . x̸=0 ∥x∥2
2. (8 points) Compute the LDLT factorization of A = 3 a 2, assuming it exists. Which
values of a and b cause (i) A to be singular? (ii) A to be positive definite?
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3. A real 3 × 3 matrix A, not necessarily symmetric, has eigenvalues 5, 3 and 0 with corresponding eigenvectors u, v and w, all unit vectors. Suppose z is a unit vector orthogonal to u and to v.
(a) (3 points) Find any solution of Ax = b if b = 2u + 3v.
(b) (5 points) Find the minimum norm least squares solution of Ax = b if b = 2u + 3v + 4z. (The answer will contain wwT somewhere in the formula. Justify your answer.)
4. (8 points) Compute the polar decompositon A = Q|A| of
9 6 6 2/3 1/318 2/3 1/3 2/3 A=6 0 12=2/3 −2/3 9 1/3 2/3 −2/3 ,
660 1/32/3
i.e., compute the matrix entries of the partial isometry Q with the same nullspace as A and the symmetric, positive semi-definite matrix |A|.
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5. (8 points) Draw the tilted ellipse (5/2)x2 +2xy+y2 = 1 and find the half-lengths of its major and minor axes from the eigenvalues of the corresponding matrix S for which (5/2)x21 + 2x1x2 + x2 = xTSx.
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