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buffordboy23
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Homework Statement
If [tex] \vec{x} [/tex] is an eigenvector of a Hermitian matrix H, let V be the set of vectors orthogonal to [tex] \vec{x} [/tex]. Show that V is a subspace, and that it is an invariant subspace of H.
The Attempt at a Solution
The Hermitian H must act on some linear space, call it K and of dimension N. This space has N linear independent vectors. As given, there exists an eigenspace with dimension 1, so V cannot have dimension greater than N-1, and as a consequence, any vector in the eigenspace cannot be a linear combination of the vectors from V. I understand that for V to be an invariant subspace, that when H acts on a vector within V, the resulting vector must always be an element of V.
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.