linear algebra proofthanks In this exercise, you will prove that commuting self-adjoint operators canbe 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 subspaceis 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 normaloperators can be simultaneously diagonalized. Of course, one needs to check thatrestriction of a normal operator to an invariant subspace is still normal.)=

a.) Let λ be an eigenvalue of T and let v be an eigenvector of T corresponding to λ, i.e. T(v) =…

Answered: In this exercise, you will prove that…

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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linear algebra proof

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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.)
=

Branch of mathematics concerned with mathematical structures that are closed under operations like addition and scalar multiplication. It is the study of linear combinations, vector spaces, lines and planes, and some mappings that are used to perform linear transformations. Linear algebra also includes vectors, matrices, and linear functions. It has many applications from mathematical physics to modern algebra and coding theory.

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