Quadratic form

A quadratic form is a homogeneous degree-two function $Q(v)$ represented in coordinates by $v^{T}Av$ with $A$ symmetric.

A quadratic homogeneous polynomial determines a unique symmetric bilinear coefficient tensor. An antisymmetric addition to its coefficient array gives zero contribution, so tensorial conditions on its coefficients must refer to this symmetric representative. In particular, the correspondence between quadratic geodesic first integrals and rank-two Killing tensors uses symmetric coefficients.

Write a real quadratic polynomial as $x^{T}Hx+2q^{T}x+c$, with $H$ symmetric. If $H$ is invertible, translation by $x_0=-H^{-1}q$ removes the linear term. An orthonormal eigenbasis then makes the remaining quadratic form diagonal. Eigenvalue signs and the translated constant determine the real geometry, such as an ellipsoid or a one-sheet hyperboloid. A singular $H$ requires separate treatment of linear terms in its null directions; a center need not exist.

Two symmetric matrices are congruent when $C=P^{T}AP$ for an invertible matrix $P$. This is the transformation of a quadratic form under an invertible linear change of coordinates. It differs from matrix similarity, which uses $P^{-1}AP$. Completing the square diagonalizes a real quadratic form by matrix congruence.

A real quadratic form $q$ is positive definite when $q(v)>0$ for every nonzero vector $v$. In a basis that diagonalizes it, this is equivalent to every diagonal coefficient being positive.

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 $a,b,c$.

Over a finite field of odd cardinality $q$, a diagonal nondegenerate ternary quadratic form $A{x}^2+B{y}^2+C{z}^2$ has a zero with $z=1$: the sets $\{A{x}^2\}$ and $\{-C-B{y}^2\}$ both have $(q+1)/2$ elements and must intersect. This gives a nonzero nonsingular zero. Over a local integer ring with these coefficients units, the Hensel lemma lifts it to an isotropic vector in the fraction field.

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 parallelogram law for a quadratic form $Q$ isConversely, under the usual divisibility assumptions, a function satisfying this identity and the appropriate homogeneity comes from a symmetric bilinear form by polarization.

The rank of a quadratic form is the rank of its associated symmetric bilinear form.

A rank-one quadratic form is the square of a linear form up to a scalar. More generally, a bilinear expression $(u^{T}x)(v^{T}y)$ is rank one because its coefficient matrix is the outer product $u{v}^T$.

The signature of a real quadratic form is the number of positive squares minus the number of negative squares in its diagonal normal form.

Over a field of characteristic other than two, a quadratic form recovers its symmetric bilinear form by

For a complex sesquilinear form $B$, the values $B(x,x)$ determine $B(x,y)$. In particular, expanding at $x+y$ and $x+iy$ shows that $B(x,x)=0$ for every $x$ implies $B=0$. Over the reals this conclusion can fail for nonsymmetric bilinear forms, as a nonzero skew-symmetric form has zero diagonal.

A symmetric bilinear form is positive semidefinite when $B(v,v)\ge 0$ for every $v$, and positive definite when equality is possible only for $v=0$.

A definite matrix is a Hermitian matrix whose associated quadratic form has a fixed sign. Positive and negative semidefinite matrices allow zero as well.

A real symmetric or complex Hermitian matrix $A$ is positive semidefinite when $x^{†}Ax\ge 0$ for every vector $x$, equivalently when all its eigenvalues are nonnegative.

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.

A real symmetric matrix admitting a finite sum of nonnegative rank-one matrices $x_j{x}_j^T$. It is both a positive semidefinite matrix and a nonnegative matrix, but these two properties alone need not imply membership in the completely positive cone in arbitrary dimension. This matrix notion is distinct from a completely positive map.

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 $P$.

For real positive semidefinite matrices $A,B$, their product need not be symmetric, but its matrix trace is nonnegative:The equality uses the cyclic property of the matrix trace, and the last matrix is a positive semidefinite matrix. Consequently the Loewner order inequality $A⪯C$ implies $\mathrm{tr}(AB)\le \mathrm{tr}(CB)$ whenever $B⪰0$.

The Loewner order on Hermitian matrices is defined by $A⪯B$ when $B-A$ is a positive semidefinite matrix.

A square root of a square matrix $A$ is a matrix $B$ satisfying $B^2=A$.

If $A=Q\mathrm{diag}({\lambda }_1,\dots ,{\lambda }_n)Q^T$ is real symmetric and positive semidefinite, its principal square root isIt is symmetric and positive semidefinite, and is positive definite exactly when $A$ is.

Every invertible real matrix has the polar decompositionwhere $P$ is symmetric positive definite and $R$ is orthogonal.

Every invertible complex matrix has the unique polar decompositionwhere $P$ is a positive-definite Hermitian matrix and $U$ is a unitary matrix. Writing $P=e^B$ gives $B=\frac{1}{2}\log (M{M}^{\ast })$, which is Hermitian.

A real symmetric matrix $A$ is positive definite whenfor every nonzero real vector $x$. It therefore defines the inner product $⟨x,y{⟩}_A=x^{T}Ay$ and is invertible.

Every real symmetric positive-definite matrix has a unique factorization $A=L{L}^T$ in which $L$ is a lower triangular matrix with positive diagonal entries.

Given a Cholesky decomposition $A=L{L}^T$, a rank-one Cholesky update computes a triangular factor of $A+x{x}^T$ in $O(n^2)$ operations without refactorizing the matrix from scratch.

A Hermitian matrix is positive definite exactly when all its leading principal minors are positive.

The condition number of a problem measures how strongly relative perturbations of its input can amplify relative changes in its output. For an invertible matrix and a chosen operator norm, $\kappa (A)=\Vert A\Vert \Vert A^{-1}\Vert$.

For a real symmetric positive-definite matrix,It measures the sensitivity of the linear system and controls convergence bounds for iterative solvers.

A complex matrix $A$ is Hermitian positive definite when $A^{\ast }=A$ and $z^{\ast }Az>0$ for every nonzero complex vector $z$.

A Toeplitz matrix is constant along each diagonal, so its entries have the form $T_{nm}=t_{n-m}$.

An $m\times m$ symmetric tridiagonal Toeplitz matrix has one constant diagonal $\alpha$ and equal constant subdiagonal and superdiagonal $\beta$. Its eigenvectors are the discrete sine transform vectors $q_j^{(k)}=\sin (jk\pi /(m+1))$, with eigenvalues $\alpha +2\beta \cos (k\pi /(m+1))$.

An $M\times M$ Toeplitz antisymmetric tridiagonal matrix with diagonal $a$, superdiagonal $b$, and subdiagonal $-b$ has eigenvaluesAll such matrices share an orthonormal eigenbasis after the same diagonal unitary change from the discrete sine basis.

A real symmetric matrix $A$ is negative definite when $x^{T}Ax<0$ for every nonzero real vector $x$. Equivalently, $-A$ is a positive-definite matrix and every eigenvalue of $A$ is negative.

Every real quadratic form is congruent to a diagonal form with entries $+1,-1,0$, and the numbers of entries of each kind are invariant.

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.

An isotropic quadratic form has a nonzero vector on which it vanishes.

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.

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 subspace $E$ is totally isotropic for $B$ when $B(u,v)=0$ for all $u,v\in E$. For a real nondegenerate bilinear form of inertia $(p,q)$, its dimension is at most $\min (p,q)$.

For a real quadratic form on an $n$-dimensional vector space, with rank $r$ and signature $s$ defined as positive minus negative index, the maximum dimension of a totally isotropic subspace is $n-(r+|s|)/2$. Quotienting by the radical of a bilinear form leaves a nondegenerate form of indices $p,q$. Projection of an isotropic subspace to each sign-coordinate space is injective, bounding its dimension by $\min (p,q)$. The radical and vectors pairing one positive with one negative unit coordinate attain the bound $n-r+\min (p,q)$.

A vector $v$ is isotropic for a quadratic form $q$ when $q(v)=0$.

For symmetric $A$, the quadratic form $x^{T}Ax$ has gradient $2Ax$.