Symmetric matrices
Sum and difference of symmetric matrices are symmetric, while product is symmetric if and only if matrices commute: $AB = BA$.
Consequence: for symmetric $A$ the $A^n$ is also symmetric.
When $A^{-1}$ exists, it is symmetric if and only if $A$ is symmetric.
Any square matrix can uniquely be written as sum of a symmetric and a skew-symmetric matrix: $X = \frac{1}{2} \left( X+X^T \right) + \frac{1}{2} \left( X-X^{T} \right)$
Real $A$ is symmetric if and only if $\langle Ax,y\rangle =\langle x,Ay\rangle \quad \forall x,y\in {\mathbb {R}}^{n}$.
Because $\langle Ax,y\rangle = (Ax)^T y = x^T A^T y = x^T Ay = <x, Ay>$.
https://math.stackexchange.com/questions/410905/symmetric-matrix-and-inner-product-langle-ah-x-rangle-langle-h-at-x-rangl
Real symmetric matrices that commute can be simultaneously diagonalized:
there exists a basis such that every element of the basis is an eigenvector for both $A$ and $B$.
https://math.stackexchange.com/questions/236212/simultaneously-diagonalizable-proof
https://math.stackexchange.com/questions/56307/simultaneous-diagonalization
After orthogonal change of basis ($CC^T=1$) symmetric matrix stays symmetric: $(C^{-1} S C)^T = C^T S^T (C^{-1})^T = C^{-1} S C$.
Orthogonal matrix.
https://en.wikipedia.org/wiki/Orthogonal_matrix
An orthogonal matrix is a square matrix whose columns and rows are orthogonal unit vectors (i.e., orthonormal vectors):
$Q^{\mathrm {T} }Q=QQ^{\mathrm {T} }=I$
${\displaystyle Q^{\mathrm {T} }=Q^{-1}}$
Orthogonal matrices preserve dot product: ${\displaystyle {\mathbf {v} }^{\mathrm {T} }{\mathbf {v} }=(Q{\mathbf {v} })^{\mathrm {T} }(Q{\mathbf {v} })={\mathbf {v} }^{\mathrm {T} }Q^{\mathrm {T} }Q{\mathbf {v} }}$