5. Let S be a summetric matrix with eigenvalues A,.. A (counted with multiplicity). Order the eigenvalues so that A,| 2 Aal 2. geglA, >0 = A=. A.. a) Show that the singular values of S are Al... In particular, rank(S) = r. b) Suppose that S= QDQ", where Q is orthogonal and D is the diagonal matrix with diagonal entries A. A. Show that S has a singular eigenvalue decomposition of the form UEQ" (i.e., V = Q). How is E related to D? How is U related to Q? e) Show that S = QDQ" is a singular value decomposition if and only if S is positive semi-definite.

5. Let S be a summetric matrix with eigenvalues A,.. A (counted with multiplicity). Order the eigenvalues so that A,| 2 Aal 2. geglA, >0 = A=. A.. a) Show that the singular values of S are Al... In particular, rank(S) = r. b) Suppose that S= QDQ", where Q is orthogonal and D is the diagonal matrix with diagonal entries A. A. Show that S has a singular eigenvalue decomposition of the form UEQ" (i.e., V = Q). How is E related to D? How is U related to Q? e) Show that S = QDQ" 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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