5. Let S be a symmetric matrix with eigenvalues 1,. .., An (counted with multiplicity). Order the eigenvalues so that |A1| > |A2| > · ·· > A,| > 0 = X,r+1 = ...= \n. a) Show that the singular values of S are |A1|,...,|A,]. In particular, rank(S) = r. b) Suppose that S = entries A1,. .. , An. QDQ", where Q is orthogonal and D is the diagonal matrix with diagonal (i) Show that S has a singular eigenvalue decomposition of the form UEQT (i.e., V = Q). (ii) How is E related to D? (iii) How is U related to Q? c) Show that S = QDQT is a singular value decomposition if and only if S is positive semi-definite.

5. Let S be a symmetric matrix with eigenvalues 1,. .., An (counted with multiplicity). Order the eigenvalues so that |A1| > |A2| > · ·· > A,| > 0 = X,r+1 = ...= \n. a) Show that the singular values of S are |A1|,...,|A,]. In particular, rank(S) = r. b) Suppose that S = entries A1,. .. , An. QDQ", where Q is orthogonal and D is the diagonal matrix with diagonal (i) Show that S has a singular eigenvalue decomposition of the form UEQT (i.e., V = Q). (ii) How is E related to D? (iii) How is U related to Q? c) Show that S = QDQT is a singular value decomposition if and only if S is positive semi-definite.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

5. Let S be a symmetric matrix with eigenvalues A1, ... , Xn (counted with multiplicity).
Order the eigenvalues so that
|A1| > |A2| > · · · > |A,| > 0 = Xr+1 = . .= \n.
a) Show that the singular values of S are |A1|, ..., |A,|. In particular, rank(S) = r.
b) Suppose that S = QDQ", where Q is orthogonal and D is the diagonal matrix with diagonal
entries A1,..., An-
(i) Show that S has a singular eigenvalue decomposition of the form UEQT (i.e., V = Q).
(ii) How is E related to D?
(iii) How is U related to Q?
c) Show that S = QDQT is a singular value decomposition if and only if S is positive semi-definite.

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