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I) If SE L(V) is an isometry on the inner product space V, then S+ il is always invertible. II) We can find a nilpotent operator N E L(R*) and v € Rª with N'v #0.. III) If N € L(V) is nilpotent then its characteristic polynomial is 2im V.

I) If SE L(V) is an isometry on the inner product space V, then S+ il is always invertible. II) We can find a nilpotent operator N E L(R*) and v € Rª with N'v #0.. III) If N € L(V) is nilpotent then its characteristic polynomial is 2im V.

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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I) If S E L(V) is an isometry on the inner product space V, then S+ il is always invertible. II) We can find a nilpotent operator NE L(R*) and v € Rª with N'v #0.. III) If N E L(V) is nilpotent then its characteristic polynomial is 2dim V. IV) There exists a normal operator T on R°, with its Euclidean inner product, which have exactly 2 eigenvalues each one of multiplicity 2. V) If T is a unitary operator on the complex inner product space, whose matrix in some basis of V has all entries real numbers, then either det T = +1 or det T = -1.
Transcribed Image Text:I) If S E L(V) is an isometry on the inner product space V, then S+ il is always invertible. II) We can find a nilpotent operator NE L(R*) and v € Rª with N'v #0.. III) If N E L(V) is nilpotent then its characteristic polynomial is 2dim V. IV) There exists a normal operator T on R°, with its Euclidean inner product, which have exactly 2 eigenvalues each one of multiplicity 2. V) If T is a unitary operator on the complex inner product space, whose matrix in some basis of V has all entries real numbers, then either det T = +1 or det T = -1.
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