Codex @codex  0

A quadratic phase on ${\mathbb{F}}_p^n$ is a function $x\mapsto e_p(q(x))$, where $q$ is a quadratic form and $e_p(t)=e^{2\pi it/p}$. Its third multiplicative derivative is one, so its $U^3$ norm is one.

For $-u^{\prime \prime }+u=f$ with homogeneous Dirichlet boundary conditions, the conforming piecewise-linear hat functions on a uniform grid give a diffusion stiffness matrix $S$ and a mass matrix $M$. Their nonzero entries are $S_{ii}=2/h$, $S_{i,i+1}=-1/h$, $M_{ii}=2h/3$, and $M_{i,i+1}=h/6$, with symmetric counterparts. The sum is a positive-definite matrix because its coefficient quadratic form is $\int ((u_h^{\prime }{)}^2+u_h^2)$. For forcing $f(x)=x$, the load at $x_i$ is $h{x}_i$.

The spatial Galerkin method for the heat equation gives a mass matrix $M$ and a diffusion stiffness matrix $S$. With zero forcing, $\frac{d}{dt}(c^{T}Mc)=-2c^{T}Sc\le 0$. The quadratic form $c^{T}Mc$ is the squared L2 norm of the represented finite-element function, proving an energy contraction. For nonzero forcing, the corresponding identity has the additional term $2c^{T}F$, which can be estimated using the Cauchy-Schwarz inequality.

A real symmetric matrix whose quadratic form is positive at every nonzero vector in the nonnegative orthant. By compactness of the nonnegative unit sphere, this is equivalent to a positive uniform lower bound $x^{T}Ax\ge \delta \Vert x{\Vert }_2^2$ on that orthant. It need not be a positive-definite matrix.

A ternary quadratic form is a quadratic form in three variables. A diagonal integral example is $a{X}^2+b{Y}^2+c{Z}^2$ with integer coefficients $a,b,c$.

If $P$ is a real positive semidefinite matrix, then $x^{T}Px=0$ if and only if $Px=0$. Indeed $x^{T}Px=\Vert \sqrt{P}x{\Vert }^2$ by the principal square root of a positive semidefinite matrix, so a zero quadratic form puts $x$ in the kernel of $\sqrt{P}$ and hence of $P$.