In this exercise, you will prove that commuting self-adjoint ope usly diagonalized. Let T, S = L(V) be commuting self-adjoint hat each eigenspace of T is S-invariant. that each eigenspace of T has an orthonormal basis consistin
In this exercise, you will prove that commuting self-adjoint ope usly diagonalized. Let T, S = L(V) be commuting self-adjoint hat each eigenspace of T is S-invariant. that each eigenspace of T has an orthonormal basis consistin
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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Transcribed Image Text:In this exercise, you will prove that commuting self-adjoint operators can
be simultaneously diagonalized. Let T, SE L(V) be commuting self-adjoint operators.
a.) Prove that each eigenspace of T is S-invariant.
b.) Deduce that each eigenspace of T has an orthonormal basis consisting of eigen-
vectors of S. (Hint: Use that the restriction of S to any S-invariant subspace
is self-adjoint; now use the spectral theorem for the restriction.)
c.) Conclude that V has an orthonormal basis consisting of simultaneous eigenvec-
tors of both T and S.
(Remark: In a similar fashion, when F C, we can prove that commuting normal
operators can be simultaneously diagonalized. Of course, one needs to check that
restriction of a normal operator to an invariant subspace is still normal.)
=
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