Eigenvalue, Quadratic-Form and Order Bounds for a Symmetric Matrix

In the setting of Symmetric and Positive Semidefinite Matrices, the Operator Norm and Spectral Notation, let $n\in\mathbb{N}$, let $A\in\mathcal{S}(n)$, and let $a,b\in\mathbb{R}$. Write $\lambda_{1}(A)\ge\dots\ge\lambda_{n}(A)$ for the ordered eigenvalues of $A$.

(i) Order and quadratic form. $aI_{n}\preceq A\preceq bI_{n}$ if and only if

$$a\lVert\xi\rVert^{2}\le\xi\cdot(A\xi)\le b\lVert\xi\rVert^{2}\qquad\text{for every }\xi\in\mathbb{R}^{n}.$$

(ii) Quadratic form and eigenvalues. $a\lVert\xi\rVert^{2}\le\xi\cdot(A\xi)\le b\lVert\xi\rVert^{2}$ for every $\xi\in\mathbb{R}^{n}$ if and only if $a\le\lambda_{n}(A)$ and $\lambda_{1}(A)\le b$.

(iii) Extreme eigenvalues as bounds. $\lambda_{n}(A)I_{n}\preceq A\preceq\lambda_{1}(A)I_{n}$.

(iv) Operator norm.

$$\lVert A\rVert=\max\bigl(|\lambda_{1}(A)|,|\lambda_{n}(A)|\bigr).$$