However, if products of three symmetric matrices are considered, any permutation is allowed.

The first fundamental form may be represented as a symmetric matrix.

In the case of the real symmetric matrix, we see that , so clearly holds.

The same is true of any a symmetric real matrix.

In particular if is a real symmetric matrix, they are the same except for transposition.

Assume for the moment that is a symmetric matrix.

Quadratic forms can be expressed as where is a symmetric matrix.

Thus is expressible as the product of two symmetric matrices.

Also if then while and again the reader may check that the product of the symmetric matrices.

He also showed, in 1829, that the eigenvalues of symmetric matrices are real.