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Let V be a finite-dimensional inner product space over F. (a) Prove that the trace of every positive operator in End(V) is non-negative. (b) Suppose T1 e End(V) and T2 e End(V) are positive operators. Prove that tr(TT) > 0.

Let V be a finite-dimensional inner product space over F. (a) Prove that the trace of every positive operator in End(V) is non-negative. (b) Suppose T1 e End(V) and T2 e End(V) are positive operators. Prove that tr(TT) > 0.

Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter4: Vector Spaces
Section4.2: Vector Spaces
Problem 38E: Determine whether the set R2 with the operations (x1,y1)+(x2,y2)=(x1x2,y1y2) and c(x1,y1)=(cx1,cy1)...
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4. Let V be a finite-dimensional inner product space over F. (a) Prove that the trace of every positive operator in End(V) is non-negative. (b) Suppose T1 e End(V) and T, e End(V) are positive operators. Prove that tr(T¡T2) > 0.
Transcribed Image Text:4. Let V be a finite-dimensional inner product space over F. (a) Prove that the trace of every positive operator in End(V) is non-negative. (b) Suppose T1 e End(V) and T, e End(V) are positive operators. Prove that tr(T¡T2) > 0.
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