面试题

A Quick Proof that Eigenvalues of a Real Symmetric Matrix are Real(ai templete Not the content of the notes will be )

A short, self-contained proof using the complex inner product -- a common interview question in linear algebra.

This is a standard result, but the proof is short enough to be worth recording.

Theorem: Eigenvalues of Real Symmetric Matrices

Let be symmetric (i.e. ). Then every eigenvalue of is real.

Proof

Let be an eigenvalue with (possibly complex) eigenvector , so . Consider the complex inner product .

Compute:

On the other hand, since is real and symmetric ():

Therefore is real. Since , we conclude is real.

Remark

The same argument, with minor modifications, shows that eigenvectors corresponding to distinct eigenvalues of a real symmetric matrix are orthogonal. This is the starting point of the spectral theorem.