(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

If [tex] \vec{x} [/tex] is an eigenvector of a Hermitian matrixH, letVbe the set of vectors orthogonal to [tex] \vec{x} [/tex]. Show thatVis a subspace, and that it is an invariant subspace ofH.

3. The attempt at a solution

The HermitianHmust act on some linear space, call itKand of dimension N. This space has N linear independent vectors. As given, there exists an eigenspace with dimension 1, soVcannot have dimension greater than N-1, and as a consequence, any vector in the eigenspace cannot be a linear combination of the vectors fromV. I understand that forVto be an invariant subspace, that whenHacts on a vector withinV, the resulting vector must always be an element ofV.

I really need help on logically connecting and formalizing these ideas into a coherent proof. Any starting points would be of great benefit, since it has been over a year since I had any linear algebra coursework. Thanks.

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# Proof: V is an invariant subspace of Hermitian H

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