21111112,2 6. (a) Let T: V→ V be a self-adjoint linear operator on a finite-dimensional realinner product space V. Show that all eigenvalues of T are nonnegative ifand only if T has a self-adjoint square root, that is, there exists a self-adjointoperator S: V→ V such that S2 = T. (Hint: spectral theorem for self-adjointoperators.)(b) Let A be a real symmetric matrix. Show that all eigenvalues of A are nonneg-ative if and only if A has a real symmetric square root, that is, there exists areal symmetric matrix B such that B2 = A.(c) Find a real symmetric square root of the matrix in Textbook Sec. 6.5 Exer-cises 2(e).

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6. (a) Let T: V → V be a self-adjoint linear operator on a finite-dimensional real inner product space V. Show that all eigenvalues of T are nonnegative if and only if T has a self-adjoint square root, that is, there exists a self-adjoint operator S: V→ V such that S² = T. (Hint: spectral theorem for self-adjoint operators.)

6. (a) Let T: V → V be a self-adjoint linear operator on a finite-dimensional real inner product space V. Show that all eigenvalues of T are nonnegative if and only if T has a self-adjoint square root, that is, there exists a self-adjoint operator S: V→ V such that S² = T. (Hint: spectral theorem for self-adjoint operators.)

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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Question

6. (a) Let T: V→ V be a self-adjoint linear operator on a finite-dimensional real
inner product space V. Show that all eigenvalues of T are nonnegative if
and only if T has a self-adjoint square root, that is, there exists a self-adjoint
operator S: V→ V such that S2 = T. (Hint: spectral theorem for self-adjoint
operators.)
(b) Let A be a real symmetric matrix. Show that all eigenvalues of A are nonneg-
ative if and only if A has a real symmetric square root, that is, there exists a
real symmetric matrix B such that B2 = A.
(c) Find a real symmetric square root of the matrix in Textbook Sec. 6.5 Exer-
cises 2(e).

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