Let T and U be a self-adjoint linear operators on an n-dimensional innerproduct space V, and let A [T], where 3 is an orthonormal basis for=V. Prove the following results.(a) T is positive definite [semidefinite] if and only if all of its eigenval-ues are positive [nonnegative].(b) T is positive definite if and only ifΣ Aijajāį > 0 for all nonzero n-tuples (a₁, a2,, an).(c) T is positive semidefinite if and only if A = B* B for some squarematrix B.

Given Data: We have to prove: (a) T is positive definite (semi definite) if and only if of eigen…

Answered: Let T and U be a self-adjoint linear…

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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Let B = {v1, V2, V3} be a basis for R3. For any u, v E R³ define || (u, v) = [u]E[v]B. • (a) Is (,) an inner product? (b) Can you find a matrix A (constructed using v1, V2, and v3) such that (u, v) = u" Av?
(3) Let V be a finite-dimensional vector space and TE L(V). Let A1,...,Am denote the distinct nonzero eigenvalues of T. Prove that T is diagonalizable if and only if dim E(A1,T) + ...+ dim E(m, T) = dim range T.
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2. Let |1), |2), |3) be an orthonormal basis for a three dimensional vector space V. Find the eigenvalues and eigenvectors of the operator A = (|1) + |2) + |3))({1| + (2| + (3) Make sure to choose the eigenvectors so that they are orthonormal.
26. Prove: If {u1, u2, . . . , u,} is an orthonormal basis for R", and if A can be expressed as A = c,u,u + c,u,u + · · + c,,4,U, then A is symmetric and has eigenvalues c1, C2, . . . , Cn .
A square matrix U E CNN is unitary if UTU = UU = I, where I is the NX N identity matrix. Prove that a square matrix is unitary if and only if the rows and columns of U form an orthonormal basis for CN.
Suppose that β and γ are ordered bases for an n-dimensional real [complex] inner product space V. Prove that if Q is an orthogonal [unitary] n ×n matrix that changes γ-coordinates into β-coordinates, then β is orthonormal if and only if γ is orthonormal.
6. Let A be an n x n hermitian matrix, i.e. A = A*. Assume that all n eigenvalues are different. Then the normalized eigenvectors { v; : j = 1,...,n} form an orthonormal basis in C". Consider B:= (Ax – ux, Ax – vx) = (Ax – ux)*(Ax – vx) | where (, ) denotes the scalar product in C" and µ,v are real constants with µ 0.
Let V be a finite-dimensional vector space over F with T in L(V,V) invertible and lambda in F \ {0}. Prove that lambda is an eigenvalue for T if and only if lambda-1 is an eigenvalue for T-1.
If b1, b2 form an orthonormal basis for C? andx is any vector in C2, then O x|4 – 2 |x · b1² |x - b2 = |x · b1|* + |x · b2* |x|4 – x - b1" – |x - be = 0 |x|4 – |x - b1 = 2 |x - b1/* |x - b2 + |x - b2 None of the given options is correct |x|4 + |x · b1 – 2 |x - bị* |x - b2 = |x - bạ| O O O O O
7. Let A be an n x n Hermitian matrix. (a) Show that (Au, v) = (u, Av) for any n-dim vectors u and v. Here (, ·) denotes the inner product. (Hint: rewrite the inner product as matrix multiplication.) (b) Let x be an eigenvector associated to the eigenvalue A. Based on part (a), show that A(x, x) = X(x, x). (c) As in part (b), show that = X; that is, the eigenvalue A is real. 1

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