Real symmetric matrix

A real symmetric matrix has real entries and satisfies $A^T=A$, so $a_{ij}=a_{ji}$. For example, $\left(\begin{array}{cc}2& 1\\ 1& 2\end{array}\right)$ has eigenvalues $3,1$ and orthogonal eigenvectors $(1,1),(1,-1)$.

The real spectral theorem says that every real symmetric matrix admits $A=QD{Q}^T$, with $Q$ an orthogonal matrix and $D$ real diagonal. Thus its eigenvalues are real and it has an orthonormal eigenbasis. The same matrix defines the quadratic form $x^{T}Ax$; its signs in eigenvector coordinates determine positive definiteness and the principal axes of a quadratic surface.