4.2d Symmetric Matrix

Symmetric Matrix

A square matrix $A\in {\mathbb{R}}^{n\times n}$ is symmetric along its diagonal if it is equal to its transpose, such that
$A=A^T.$
The number of free entries (entries above and on the diagonal) in a symmetric matrix is $\frac{n(n+1)}{2}$.

Properties

Here we must define a few properties of a real symmetric matrix. If a matrix $A\in {\mathbb{R}}^{n\times n}$ is real and symmetric, then:

The eigenvalues of A are always real.

$A$ is always diagonalizable.

The eigenvectors of A form an orthonormal basis.