Suppose TEL(V) has singular value decomposition given byTv = $₁(v, e₁) f₁ +... + Sn (v, en) fnfor every ve V, where $₁...., Sn are the singular values of T ande.....en and fi..... fn are orthonormal bases of V.(d) Suppose T is invertible. Prove that if v € V, then(v. fn)enT-¹y =(v. file₁$1++Snfor every v € V.

The problem is solved using the concept of singular value decomposition.

Answered: Suppose TEL(V) has singular value…

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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5. Let S be a summetric matrix with eigenvalues A,...A. (counted with multiplicity). Order the eigenvalues so that |2 geglA,| > 0 = A,+1 == A.. a) Show that the singular values of S are Al,.A,. 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" (ie., 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. %3D
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In the vector space P of polynomials of 2' degree atmost two, the first column of the matrix of transition from the basis B=(e₁=1+x;e₂=x+x²e₂=3+x²) to basis A={r= 1− 3x; s=x; t=x²} is: A. 1 O*:) B. -2 *(-9 - 1 1 06 (8) D. 0° (:)
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12. Let T be a linear operator on a finite-dimensional vector space V, and let A be an eigenvalue of T with corresponding eigenspace and generalized eigenspace Ex and KA, respectively. Let U be an invertible linear opera- tor on V that commutes with T (i.e., TU = UT). Prove that U(Ex) = Ex and U(K) K₁. =
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Let n ≥ 2. Consider the reversing operator R: R" → Rn defined by R(a₁, 02, ..., an-1, an) = (an, an-1,..., a2, 0₁). We denote by E₁ and E-1 the eigenspaces of eigenvalues X = 1 and λ = -1, respec- tively. (1) (2) If n is even, find a basis of E₁ and a basis of E-1, and determine whether or not R is diagonalizable. If n is odd, find a basis of E₁ and a basis of E-1, and determine whether or not R is diagonalizable.
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Theorem 3. Let A be a bounded linear operator acting between two real Hilbert spaces. Then 1. [R(A)] = Nr(A*). 2. R(A) = [N(A*)]*. 3. [R(A*)]+ = N(A). 4. R(A*) = [N(A)]+. Proof. Part 1 is just Theorem 1. To prove part 2, take the orthogonal complement of both sides of 1 obtaining [R(A)] = [N(A*)]*. Since R(A) is a subspace, the result follows. Parts 3 and 4 are obtained from 1 and 2 by use of the relation A** = A. I

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