et S be a symmetric matrix with eigenvalues A1,..., An (counted with multiplicity). Drder the eigenvalues so that |Ai| 2 |A2| 2 ·. A,| > 0 = Ap4+1 = • ·· = \n. ) Show that the singular values of S are |A1|, ...,|A,|. In particular, rank(S) = r. ) 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 UEQ" (i.e., V = Q). (ii) How is E related to D? (iii) How is U related to Q? ) Show that S = QDQ" is a singular value decomposition if and only if S is positive semi-definite.

et S be a symmetric matrix with eigenvalues A1,..., An (counted with multiplicity). Drder the eigenvalues so that |Ai| 2 |A2| 2 ·. A,| > 0 = Ap4+1 = • ·· = \n. ) Show that the singular values of S are |A1|, ...,|A,|. In particular, rank(S) = r. ) 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 UEQ" (i.e., V = Q). (ii) How is E related to D? (iii) How is U related to Q? ) 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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