Symmetric Matrices

A square matrix

is called a symmetric matrix if

. It has many nice properties that can be summarized into the following theorem: Spectral Theorem for symmetric matrices: For any

symmetric matrix

, we have

has real eigenvalues (counting multiplicities).
Any two eigenvectors corresponding to two distinct eigenvalues are orthogonal.
The dimension of each eigenspace equals the multiplicity of the corresponding eigenvalue.
is diagonalizable. Moreover, we can choose an orthogonal matrix such that , where is a diagonal matrix.

The proof of spectral theorem is beyond the scope of this course. We can only prove (2) as follows: Suppose

and

are eigenvectors of

corresponding to eigenvalues

and

respectively. And we assume

. Then we have the following:

Since

, which implies that

and

are orthogonal.

In the applet below, you can visualize the diagonalization of a

symmetric matrix.

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