Symmetric matrices

A square matrix $$A \in \mathbb{R}^{n\times n}$$ is called symmetric if $$A^T = A$$.

Properties of Symmetric Matrices

 * The eigenvalues of a symmetric matrix are real.
 * There is a set of orthonormal eigenvectors of $$A$$, i.e., $$\exists q_1, \dots, q_n $$ s.t. $$A q_i = \lambda_i q_i$$ and $$q_i^t q_j = \delta_{ij}$$ where $$\delta_{ij}$$ is Kronecker's delta. In matrix form, there exists a matrix $$Q$$ s.t.:

$$Q^T A Q_T = \Lambda$$

What's the point of Symmetric matrices?
Symmetric matrices are worth studying for a number of reasons. Firstly, when working with quadratic forms: one can always find a symmetric equivalent of a quadratic form of a matrix (see Quadratic form), and it's advantageous to work on the symmetric one because it's easy to find the minimum and maximum of the quadratic form via eigenvalues (see Quadratic form). A related property is that the positive definiteness of a symmetric matrix can be determined from its eigenvalues (see Positive definite matrix).

The property listed above ($$Q^T A Q_T = \Lambda$$) is also a very plausible one for symmetric matrices as it leads to a very intuitive decomposition.