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
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ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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5. Let S be a symmetric matrix with eigenvalues A1,..., An (counted with multiplicity).
Order the eigenvalues so that
|A:| 2 |A2| > · .> |A,| > 0 = A,+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,..., dn.
(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?
c) Show that S = QDQ™ is a singular value decomposition if and only if S is positive semi-definite.
%3D
Transcribed Image Text:5. Let S be a symmetric matrix with eigenvalues A1,..., An (counted with multiplicity). Order the eigenvalues so that |A:| 2 |A2| > · .> |A,| > 0 = A,+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,..., dn. (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? c) Show that S = QDQ™ is a singular value decomposition if and only if S is positive semi-definite. %3D
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