Properties of the Norm of a Symmetric Real Matrix

Let $n$ be a natural number with $1\le n$ and let $\mathbb{R}$ be the real numbers with the order $\le$ of its ordered field structure and the absolute value $|\cdot|$. Let $A$ and $B$ be symmetric real $n\times n$ matrices and let $\mu,\lambda\in\mathbb{R}$. Write $I_n$ for the identity matrix of size $n$, $0_n$ for the real $n\times n$ matrix all of whose entries are $0$, $A+B$ for the sum and $\mu A$ for the scalar multiple of matrices, and set $-\lambda I_n=(-\lambda)I_n$; the matrices $I_n$ and $0_n$ are symmetric, their entries being unchanged when the two indices are interchanged, and hence $A+B$, $\mu A$, $\lambda I_n$ and $-\lambda I_n$ are symmetric by claim 1 of The Positive Semidefinite Ordering is a Partial Order Compatible with the Linear Structure. Write $P\preceq Q$ for the positive semidefinite ordering.

On Euclidean space $\mathbb{R}^n$, a real vector space by Euclidean Space $\mathbb{R}^n$ is a Real Vector Space, write $\xi\cdot\zeta$ for the dot product, $\lVert\xi\rVert$ for the Euclidean norm and $A\xi$ for the matrix-vector product, and abbreviate $\lVert\xi\rVert^{2}=\lVert\xi\rVert\cdot\lVert\xi\rVert$. Let $\lVert A\rVert$ denote the norm of the symmetric matrix $A$.

Then the following hold.

1. (Nonnegativity) $0\le\lVert A\rVert$.

2. (Quadratic form bound) $|\xi\cdot(A\xi)|\le\lVert A\rVert\,\lVert\xi\rVert^{2}$ for every $\xi\in\mathbb{R}^n$.

3. (Characterization by the semidefinite ordering) If $0\le\lambda$, then $\lVert A\rVert\le\lambda$ holds if and only if both $-\lambda I_n\preceq A$ and $A\preceq\lambda I_n$ hold. In particular $-\lVert A\rVert I_n\preceq A$ and $A\preceq\lVert A\rVert I_n$.

4. (Definiteness) $\lVert A\rVert=0$ if and only if $A=0_n$.

5. (Absolute homogeneity and triangle inequality) $\lVert\mu A\rVert=|\mu|\,\lVert A\rVert$ and $\lVert A+B\rVert\le\lVert A\rVert+\lVert B\rVert$.