Proof. If is an eigenvalue and v a corresponding eigenvector of an symmetric matrix A.
Then . If we multiply each side of this equation on the left by , then we obtain
Thus, is a real number. To further verify, you apply a conjugate transpose and
, since A is a symmetric matrix.
Lemma 2. Let A be a real symmetric matrix. There is an orthogonal matrix such that is diagonal.
Proof. We first show that eigenvectors from different eigenspaces with respect to a symmetric matrix A are orthogonal.
Let and be eigenvectors corresponding to distinct real eigenvalues and of the matrix A. We shall show that . Since A is a symmetric matrix, . This implies that Since and are distinct by hypothesis, .
The above showed that eigenvectors from different eigenspaces are orthogonal. By applying a Gram-Schmit process, eigenvectors obtained within same eigenspaces become orthonormal. This ensures that there exists an orthogonal matrix whose columns are normalized eigenvectors of a symmetric matrix A.
be an orthogonal matrix whose columns are eigenvectors of A.
Thus, and , since U is an orthogonal matrix
To apply an induction, assume symmetric matrix A holds the above lemma and show symmetric matrix holds the above lemma as well. The construction shown in the lemma 2 can be used without much modification.