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. $$Q^T A Q = \Lambda$$, and therefore $$A = Q\Lambda Q^T$$, where $$\Lambda$$ is the diagonal matrix of eigenvalues of $$A$$.

Two Interpretations
The last property above is worth investigating further. Saying that $$A = Q\Lambda Q^T$$ is also saying that one can obtain $$y = Ax$$ via the following block diagram



This block diagram leads to the following interpretation for the linear mapping $$Ax$$:
 * 1) Rotate $$x$$ by $$Q^T$$
 * 2) Diagonal real scale ('dilation') by $$\Lambda$$
 * 3) Rotate back by $$Q$$.

Another interpretation is a linear combination of 1-dimensional projections:

$$A = \sum_{i=1}^n \lambda_i q_i q_i^T$$

The interpretation of the latter is illustrated in the example figure below:



According to this interpretation, the matrix-vector multiplication $$Ax$$ can be seen as:


 * 1) Resolving $$x$$ into its $$q_i q_i^T$$ components
 * 2) Scaling each component by the corresponding eigenvalue $$\lambda_i$$
 * 3) Reconstituting with basis $$q_i$$.

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 ($$Q^T A Q = \Lambda$$) is also a very plausible one for symmetric matrices as it leads to very intuitive decompositions.

Symmetric and positive definite matrices define ellipsoids.