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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.

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.

Advanced Engineering Mathematics
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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6（b) prove it in two directions

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).
Transcribed Image Text: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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