Let be a matrix and . Let be an orthonormal basis for consisting of eigenvectors for , where is a self-adjoint linear transformation on .

Prove that is an eigenvalue for and is an orthonormal basis for .

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- Jan 25th 2011, 01:03 AMproblemOrthonormal Basis
Let be a matrix and . Let be an orthonormal basis for consisting of eigenvectors for , where is a self-adjoint linear transformation on .

Prove that is an eigenvalue for and is an orthonormal basis for . - Jan 25th 2011, 06:00 AMHallsofIvy
Since we are given that these vectors are an orthonormal basis for and we want to prove it is an orthonormal basis for , it would appear that we just want to prove that . Self adjoint transformations have several nice properities that you would want to prove first, if you haven't already:

All eigenvalues are real.

If u and v are eigenvectors corresponding to distinct eigenvalues, then u and v are orthogonal.

If are eigenvectors and we restrict the transformation to the**orthogonal complement**of the span of the restriction is still self adjoint.

That way, we can start with one eigenvector, the restict ourselves to the orthogonal complement of its span, find a second eigenvector, etc. That way you should be able to show that there are, in fact, n independent eigenvectors and so