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For three vectors $a,b,c$ in an inner-product space, expansion of each squared norm provesTherefore the second-order backward differentiation formula $3U^{n+2}-4U^{n+1}+U^n=2kL{U}^{n+2}$ with a dissipative operator $L$ decreases the discrete energy ${\mathcal{E}}_n=\Vert U^{n+1}{\Vert }^2+\Vert 2U^{n+1}-U^n{\Vert }^2$. This quadratic form is equivalent to the product norm of the two time levels with constants independent of $k$ and $L$. It proves uniform stability even when amplification roots of a polynomial coalesce inside the unit disk, where an eigenvector-separation proof may lose its bound.

Optimization of a quadratic form over binary choices, here using signs $x_i\in \{-1,1\}$. Lifting $x{x}^T$ turns the objective into a matrix trace; dropping the rank-one condition gives a semidefinite relaxation of binary quadratic optimization.

For positive block sizes, the real matrix $\left(\begin{array}{cc}aJ& bJ\\ bJ& aJ\end{array}\right)$ is a positive semidefinite matrix exactly when $a\ge |b|$. For test-vector block sums $s,t$, its quadratic form is $(a+b)(s+t{)}^2/2+(a-b)(s-t{)}^2/2$. This proves sufficiency; testing $s=\pm t$ proves necessity. Zero sums in both blocks describe part of its kernel.

If an elliptic curve over $\mathbb{Q}$ has Mordell-Weil group of rank $r$, then its canonical height of an elliptic curve is a positive-definite quadratic form on the free lattice. Consequently

A real symmetric matrix $A$ is copositive if its quadratic form satisfies $x^{T}Ax\ge 0$ for every $x$ in the nonnegative orthant. Every positive semidefinite matrix and every symmetric nonnegative matrix is copositive, as is their sum. The converse fails for the Horn copositive matrix.

A vector $v$ can be included in a diagonal basis for a real quadratic form precisely when $v\ne 0$ and either its quadratic value is nonzero or it lies in the radical of a bilinear form. A nonzero quadratic value gives the direct splitting $\mathbb{R}v\oplus v^{\perp }$ and induction. A nonzero radical vector can be included in a radical basis. Conversely, an isotropic vector in a diagonal basis is orthogonal to every basis vector, and must lie in the radical.

On the product of an elliptic curve with itself, let $D_{+}$ and $D_{-}$ be the diagonal and the graph of negation. The principal divisor of $x(P)-x(Q)$ is $D_{+}+D_{-}-2(\{O\}\times E)-2(E\times \{O\})$: equality of the coordinates means $Q=\pm P$, while $x$ has a double pole at $O$. Pulling back along $(\varphi ,\psi )$ and taking degrees proves the parallelogram law for the degree of an isogeny when $\varphi ,\psi$ are nonzero and $\varphi \ne \pm \psi$. The exceptional cases follow from $\mathrm{deg}[2]=4$. The same argument uses the quotient by negation in characteristic two. Thus degree, with degree zero assigned to the zero map, is a positive quadratic form on $\mathrm{Hom}(E_1,E_2)$.

A triangular taper majorizes a fixed fraction of the summation interval. Its Fourier kernel is the Fejér kernel, bounded by $C\min (m,(m\Vert \theta {\Vert }^2{)}^{-1})$. For separated sample points, split each row sum at distance $1/m$: the plateau and square-decay tail each contribute $O({\delta }^{-1})$, while the diagonal contributes $O(m)$. Bounding the resulting quadratic form proves the dual sieve with constant $C(N+{\delta }^{-1})$. Operator norm duality gives the primal sieve.

A Feynman-Kitaev Hamiltonian penalizes incorrect input initialization, disagreement between successive quantum circuit steps and clock labels, and optionally a rejecting output. Its propagation quadratic form is a sum of $\frac{1}{2}\Vert {\eta }_t-U_t{\eta }_{t-1}{\Vert }^2$. Without output penalty, its zero-energy space consists of correctly initialized computational history states. A nonlocal quantum clock makes the propagation formula simple but does not supply fixed qubit locality by itself.

The fixed coordinate subspace of dimension $k$ in ${\mathbb{R}}^p$ consists of vectors whose coordinates after the first $k$ vanish. Restricting a quadratic form to it uses only the leading $k$ by $k$ matrix block. It specifies one fixed support; allowing every support of size $k$ would introduce another combinatorial counting step.

For a real symmetric positive-definite three-dimensional matrix, the reciprocal quadratic form is locally integrable and defines a regular tempered distribution. With the Fourier kernel $e^{-ix\cdot \xi }$, radial Abel regularization gives $\mathcal{F}(|x{|}^{-2})=2{\pi }^2/|\xi |$. The linear change of variables $y=G^{1/2}x$ gives the displayed formula. No principal value or origin-supported correction is required. The result is homogeneous of degree $-1$ on the frequency side.

For complex matrices, the inner product $⟨A,B{⟩}_F=\mathrm{tr}(A^{\ast }B)$ is conjugate-linear in its first argument. On real symmetric matrices it becomes $\mathrm{tr}(AB)$. In particular $⟨A,x{x}^T{⟩}_F=x^{T}Ax$, identifying quadratic forms with linear tests on matrices.

For a matrix $G$ with independent standard normal entries, $G^{T}G/n$ is a Gaussian empirical Gram matrix and has expectation equal to the identity. With mean known to be zero it is the raw empirical second moment. Subtracting a sample mean would give a different estimator. On any fixed unit vector its quadratic form is a chi-squared variable divided by $n$.

For a nondegenerate diagonal quadratic form $q=⟨a_1,\dots ,a_m⟩$, this product of quadratic Hilbert symbols is independent of diagonalization. Over a p-adic field, dimension, determinant square class and this invariant classify the form up to isometry. The product of the local invariants of a globally diagonalized form is one by the Hilbert reciprocity law; real places additionally record signature.

A nondegenerate quadratic form over a number field is isotropic if and only if it is isotropic over every completion. Over $\mathbb{Q}$, these are the real field and all P-adic numbers. It turns a rational existence problem into local tests; denominators can then be cleared for an integral homogeneous equation.

The five-dimensional Horn copositive matrix has diagonal entries $1$, entries $-1$ on the edges of the five-cycle, and entries $1$ on the remaining pairs:For $x\ge 0$, a cyclic relabelling puts a smallest coordinate at $x_5$. The identitythen proves that $H$ is a copositive matrix.

However, $H$ is outside the positive-semidefinite-plus-nonnegative cone. Set $w=(1,2,1,0,0{)}^T$. Its quadratic form is zero. If $H=P+N$ with $P$ a positive semidefinite matrix and $N$ a symmetric nonnegative matrix, both $w^{T}Pw$ and $w^{T}Nw$ must vanish. Positivity of the first three coordinates of $w$ forces every entry of the leading $3\times 3$ block of $N$ to vanish. Applying the argument to all cyclic shifts of $w$ forces every entry of $N$ to vanish, since each pair of indices lies in a cyclic interval of length three. This would make $H$ a positive semidefinite matrix, but zero quadratic form of a positive semidefinite matrix would then give $Hw=0$, whereas $Hw=(0,0,0,2,2{)}^T$.

A hyperbolic plane is a two-dimensional quadratic space with a basis $e,f$ satisfying $q(e)=q(f)=0$ and $B(e,f)=1$. Its diagonal form has one positive and one negative square, so its signature is zero.

The interior consists exactly of strictly copositive matrices. If $x^{T}Ax$ has positive minimum $\delta$ on the compact nonnegative unit sphere, perturbations of operator norm less than $\delta$ preserve this positive lower bound there. Conversely, if a unit nonnegative vector has zero quadratic form, $A-\epsilon I$ leaves the copositive cone for every $\epsilon >0$. Thus such a matrix is not interior. The identity matrix is a convenient interior point.

For a real quadratic form $Q(v)=v^{T}Av$ with $A$ a symmetric matrix, its stabilizer is $G_Q=\{g\in {\mathrm{GL}}_n(\mathbb{R}):g^{T}Ag=A\}$. Its Lie algebra consists exactly of $X$ satisfying $X^{T}A+AX=0$. Differentiation proves necessity; the matrix exponential proves sufficiency because $e^{t{X}^T}A{e}^{tX}=A$. The result holds even when the quadratic form is degenerate.