Norm of a Symmetric Real Matrix

Let $n$ be a natural number with $1\le n$ and let $A$ be a symmetric real $n\times n$ matrix. On Euclidean space $\mathbb{R}^n$ write $\xi\cdot\zeta$ for the dot product, $\lVert\xi\rVert$ for the Euclidean norm of a point $\xi$, and $A\xi$ for the matrix-vector product; let $|\cdot|$ be the absolute value on the real numbers, ordered by $\le$ as an ordered field.

The norm of $A$, written $\lVert A\rVert$, is the least upper bound, which exists by The Quadratic Form of a Real Square Matrix is Bounded on the Closed Unit Ball, of the set

$$\bigl\{\,|\xi\cdot(A\xi)|\ :\ \xi\in\mathbb{R}^n,\ \lVert\xi\rVert\le1\,\bigr\}.$$

The symbol $\lVert\,\cdot\,\rVert$ is used both for the Euclidean norm of a point of $\mathbb{R}^n$ and for the norm of a symmetric real matrix; the argument determines which is meant.