Let T be self-adjoint on a finite-dimensional inner product space V. Prove thatVæ E V, ||T(x)± ix||? = ||T(x)||² + ||x||?.Hence prove that T – il is invertible and that [(T – il))* = (T + iI)-1.

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Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.5: Permutations And Inverses

Problem 10E: 10. Let and be mappings from to. Prove that if is invertible, then is onto and is...

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Let T be self-adjoint on a finite-dimensional inner product space V. Prove that Vr E V, ||T(x)± i||? = ||T (x)||² + ||æ||?. Hence prove that T – iI is invertible and that [(T – il)-']* = (T + iI)-!.