4. Show that for any bounded linear operator T on H, the operators T₁=1/√(T+T*) and T₁=1 (1 == (T-T*) 2i are self-adjoint. Show that T = T₁+iT₂, T* = T₁-iT₂. Show uniqueness, that is, T₁+iT₂ = S₁+iS₂ implies S₁ = T₁ and S₂ T₂; here, S₁ and S₂ are self-adjoint by assumption.

4. Show that for any bounded linear operator T on H, the operators T₁=1/√(T+T) and T₁=1 (1 == (T-T) 2i are self-adjoint. Show that T = T₁+iT₂, T* = T₁-iT₂. Show uniqueness, that is, T₁+iT₂ = S₁+iS₂ implies S₁ = T₁ and S₂ T₂; here, S₁ and S₂ are self-adjoint by assumption.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter7: Distance And Approximation

Section7.4: The Singular Value Decomposition

Problem 54EQ

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