5. Let S be a symmetric matrix with eigenvalues A1, ., An (counted with multiplicity). Order the eigenvalues so that |A1| > |A2| > · · ·> A,| > 0 = X,+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. 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 diagonalentries 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.

It is given that Let S be a symmetric matrix with eigenvalues A1, ., An (counted with…

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Advanced Engineering Mathematics

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ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

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 |A1| > |A2| > · · ·> A,| > 0 = X,+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.

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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