Perturbation Splitting of a Quadratic Form

Let $n\ge1$ be a natural number and let $\mathbb{R}$ be the real numbers with the order $\le$ of their ordered field structure. On Euclidean space $\mathbb{R}^{n}$, a real vector space, write $z+z'$ for the sum of points, $z-z'$ and $z\cdot z'$ for the difference and dot product, and $\lVert\,\cdot\,\rVert$ for the Euclidean norm, abbreviating $\lVert z\rVert^{2}=\lVert z\rVert\cdot\lVert z\rVert$.

Let $A$ be a symmetric real $n\times n$ matrix, write $Az$ for the matrix-vector product, $A^{2}$ for the product $AA$, $P+Q$ for the sum of real matrices, $\mu P$ for the scalar multiple, and $\lVert A\rVert$ for the norm of the symmetric matrix $A$. Let $\varepsilon\in\mathbb{R}$ satisfy $0<\varepsilon$, so that $\varepsilon^{-1}$ exists by claim 7 of Elementary Order Arithmetic in an Ordered Field.

Then the following hold.

1. (Symmetry of the associated bilinear form) For all $w,z\in\mathbb{R}^{n}$,

$$w\cdot(Az)=z\cdot(Aw).$$

2. (Quadratic form of the square) For every $\xi\in\mathbb{R}^{n}$,

$$\xi\cdot(A^{2}\xi)=\lVert A\xi\rVert^{2}.$$

3. (Perturbation splitting) For all $x,\xi\in\mathbb{R}^{n}$,

$$x\cdot(Ax)\;\le\;\xi\cdot\bigl((A+\varepsilon A^{2})\xi\bigr)+\bigl(\varepsilon^{-1}+\lVert A\rVert\bigr)\,\lVert x-\xi\rVert^{2}.$$