Vector, Entry and Comparison Bounds for the Norm of a Symmetric Real Matrix

We work in the setting of Euclidean Space and Real Matrices: Standing Notation, whose notation is fixed for every dimension and is used here with a natural number $n$ satisfying $1\le n$: the real numbers, the absolute value $|\cdot|$, the initial segments $[n]$, and Euclidean space $\mathbb{R}^{n}$ with its sum and difference of points, dot product and Euclidean norm $\lVert\,\cdot\,\rVert$, the real matrices $\mathcal{M}_{n}(\mathbb{R})$, their sums, differences and scalar multiples, the product, the transpose, the matrix-vector product, the identity matrix $I_{n}$, the zero matrix $0_{n}$ and the square $A^{2}$, the set $\mathcal{S}(n)$ of symmetric real $n\times n$ matrices, which contains $aI_{n}$ for every real $a$, the norm $\lVert A\rVert$ of a symmetric real matrix and the standard basis vectors $e_{i}$, for which the $i$th coordinate of a point $z$ equals $z\cdot e_{i}$, are all as fixed there.

Let $A\in\mathcal{M}_{n}(\mathbb{R})$, let $X\in\mathcal{S}(n)$, let $z,x,y\in\mathbb{R}^{n}$, let $a\in\mathbb{R}$ and let $i,j\in[n]$. Then the following hold.

1. (Entries from the standard basis) (i) $A_{ij}=e_{i}\cdot(Ae_{j})$.

2. (Squares of symmetric matrices) (ii) The matrix $X^{2}$ lies in $\mathcal{S}(n)$, and

$$\lVert Xz\rVert^{2}=z\cdot(X^{2}z).$$

3. (Bound on the image of a vector) (iii) $\lVert Xz\rVert\le\lVert X\rVert\,\lVert z\rVert$.

4. (Bound on the entries) (iv) $|X_{ij}|\le\lVert X\rVert$.

5. (Bound by the entries) (v) $\displaystyle\lVert X\rVert\le\sum_{i=1}^{n}\sum_{j=1}^{n}|X_{ij}|$.

6. (Multiples of the identity) (vi) The matrix $aI_{n}$ lies in $\mathcal{S}(n)$ and satisfies $z\cdot\bigl((aI_{n})z\bigr)=a\,\lVert z\rVert^{2}$; moreover $\lVert aI_{n}\rVert=|a|$.

7. (Comparison of quadratic forms) (vii) $\bigl|x\cdot(Xx)-y\cdot(Xy)\bigr|\le\lVert X\rVert\,\lVert x-y\rVert\,\lVert x+y\rVert$.