Quadratic forms are the degree-two part of algebra. They appear when a function is approximated to second order, when a symmetric matrix is studied through its associated scalar expression, and when a positive definite form supplies the square of a length. General quadratic forms are broader than metric geometry: they may be indefinite or degenerate, and their zero sets can contain nonzero vectors. In linear algebra they translate symmetric matrices into geometry; in algebra they encode bilinear data with arithmetic content; in analysis they underlie energy functionals and elliptic estimates. The same object connects [bilinear forms](/page/Bilinear%20Form), [inner product spaces](/page/Inner%20Product%20Space), [symmetric matrices](/page/Symmetric%20Matrix), [eigenvalues](/page/Eigenvalue), and [quadratic polynomials](/page/Polynomial). A first way to meet a quadratic form is through real coordinates. If $A$ is a symmetric matrix, then $x \mapsto x^\top A x$ is a scalar quantity built from all pairwise products $x_i x_j$. Symmetry matters because the expression only sees the symmetric part of a matrix: the antisymmetric part vanishes when placed between $x^\top$ and $x$. ## Definition The core notion should not depend on a preferred coordinate system. A quadratic form is the kind of scalar-valued function whose value scales like a square and whose mixed second-order part is bilinear. Over fields of characteristic not equal to $2$, this gives a clean intrinsic definition and keeps the familiar matrix formula as a coordinate representation rather than the definition itself. [definition: Quadratic Form] Let $F$ be a field with $\operatorname{char}(F) \ne 2$, and let $V$ be a [vector space](/page/Vector%20Space) over $F$. A function $q: V \to F$ is a quadratic form if $q(av)=a^2q(v)$ for all $a \in F$ and $v \in V$, and the function $B_q: V \times V \to F$ defined by \begin{align*} B_q(u,v)=\frac{1}{2}\bigl(q(u+v)-q(u)-q(v)\bigr) \end{align*} is a symmetric bilinear form. [/definition] This definition makes the parent concept independent of coordinates. It also names the associated symmetric bilinear form $B_q$, which is the object used to define radicals, nondegeneracy, and matrix representatives later on. The most concrete way to compute with the definition is to choose real coordinates. This is the form that appears in multivariable calculus, optimization, spectral theory, and the classification of conic sections. It packages a homogeneous degree-two polynomial in $n$ real variables as a matrix expression. [definition: Real Quadratic Form] A real quadratic form on $\mathbb{R}^n$ is a function $Q: \mathbb{R}^n \to \mathbb{R}$ for which there exists a real symmetric matrix $A \in \mathbb{R}^{n \times n}$ such that \begin{align*} Q(x) = x^\top A x \end{align*} for every $x \in \mathbb{R}^n$. [/definition] Writing $x = (x_1, \ldots, x_n)$ reveals the polynomial hidden inside the matrix formula: \begin{align*} x^\top A x = \sum_{i=1}^n \sum_{j=1}^n A_{ij}x_i x_j. \end{align*} If $A$ is symmetric, the off-diagonal terms occur in matched pairs $A_{ij}x_i x_j$ and $A_{ji}x_jx_i$. Matrix formulas are excellent for calculation, but they hide the invariant source of many quadratic forms. In geometry and functional analysis, the primary object is often a bilinear pairing that measures how two vectors interact; energies, squared lengths, and second variations are then obtained by feeding the same vector into both slots. To compare that two-vector data with a one-vector quadratic quantity, we need a named construction that records precisely the diagonal part of a bilinear form. [definition: Quadratic Form Associated to a Bilinear Form] Let $F$ be a field, let $V$ be a vector space over $F$, and let $B: V \times V \to F$ be a bilinear form. The quadratic form associated to $B$ is the function $q_B: V \to F$ defined by \begin{align*} q_B(v) = B(v,v). \end{align*} [/definition] This construction separates two pieces of structure: the bilinear form remembers interactions between two possibly different vectors, while the quadratic form records only the diagonal values. Over fields of characteristic not equal to $2$, the diagonal function depends only on the symmetric part of $B$, and symmetric bilinear forms can be recovered from their quadratic forms by polarization. In characteristic $2$, this relationship changes in a serious way, so the distinction is not cosmetic. Many algebraic questions require allowing coefficients in rings rather than fields. For example, integral quadratic forms remember divisibility and congruence information that disappears after extending scalars to $\mathbb{Q}$ or $\mathbb{R}$. The polynomial definition is the most stable way to state the concept over a commutative ring. [definition: Quadratic Form Over a Commutative Ring] Let $R$ be a commutative ring. A quadratic form in $n$ variables over $R$ is a map $q: R^n \to R$ together with a homogeneous coordinate expression \begin{align*} q(x_1, \ldots, x_n)=\sum_{1 \le i \le j \le n} a_{ij}x_i x_j \end{align*} for some coefficients $a_{ij} \in R$, such that $q(x_1,\ldots,x_n)$ is evaluated by this expression for every $(x_1,\ldots,x_n) \in R^n$. [/definition] The ring definition keeps both pieces visible: the graph-context object is a function $q: R^n \to R$, while the homogeneous expression records the coefficients that make it a quadratic form. Over a general ring distinct polynomial expressions can define the same function on the underlying set, and not every operation involving division by $2$ is available. The coefficient expression is therefore part of the data used in algebraic and arithmetic questions, not merely a disposable way to evaluate the function. Over rings, the same polynomial can carry arithmetic data that is invisible over the [real numbers](/page/Real%20Numbers). Integral quadratic forms are central in number theory because they ask which integers can be represented by a fixed [homogeneous polynomial](/page/Homogeneous%20Polynomial). [example: Integral Binary Quadratic Form] Let $q: \mathbb{Z}^2 \to \mathbb{Z}$ be defined by \begin{align*} q(x,y)=x^2+xy+y^2. \end{align*} This is the function induced by a quadratic form over the commutative ring $\mathbb{Z}$, since the expression is homogeneous of degree $2$ with integer coefficients. It represents $1$ because \begin{align*} q(1,0)=1^2+1\cdot 0+0^2=1, \end{align*} and it represents $3$ because \begin{align*} q(1,1)=1^2+1\cdot 1+1^2=3. \end{align*} We show that $q$ does not represent $2$ over $\mathbb{Z}^2$. Suppose, for contradiction, that $x,y \in \mathbb{Z}$ satisfy $q(x,y)=2$. Then \begin{align*} x^2+xy+y^2=2. \end{align*} Multiplying by $4$ gives \begin{align*} 4x^2+4xy+4y^2=8. \end{align*} Since \begin{align*} (2x+y)^2=4x^2+4xy+y^2, \end{align*} we have \begin{align*} 4x^2+4xy+4y^2=(2x+y)^2+3y^2. \end{align*} Therefore \begin{align*} (2x+y)^2+3y^2=8. \end{align*} The term $(2x+y)^2$ is nonnegative, so $3y^2 \le 8$. Hence $y^2 \le 2$, and because $y \in \mathbb{Z}$ this forces $y \in \{-1,0,1\}$. If $y=0$, then \begin{align*} (2x+y)^2=8, \end{align*} which is impossible because $2x+y$ is an integer and no integer square equals $8$. If $y=1$ or $y=-1$, then $3y^2=3$, so \begin{align*} (2x+y)^2=8-3=5, \end{align*} which is impossible because no integer square equals $5$. Thus no pair $(x,y) \in \mathbb{Z}^2$ satisfies $q(x,y)=2$. This example illustrates the arithmetic question attached to an integral form: determine the set of values $q(x,y)$ as $(x,y)$ ranges through $\mathbb{Z}^2$. [/example] When a vector space has a fixed ordered basis, a quadratic form becomes computable through its coordinate column. This motivates attaching a matrix to the form, because determinants, rank, eigenvalues, and change-of-basis formulas all require a matrix representative. [definition: Matrix of a Quadratic Form] Let $V$ be a finite-dimensional vector space over a field $F$ with $\operatorname{char}(F) \ne 2$, let $\mathcal B = (e_1, \ldots, e_n)$ be an ordered basis of $V$, and let $q: V \to F$ be a quadratic form. The matrix of $q$ in the basis $\mathcal B$ is the symmetric matrix $A \in F^{n \times n}$ for which \begin{align*} q(v) = [v]_{\mathcal B}^\top A [v]_{\mathcal B} \end{align*} for every $v \in V$. [/definition] A [change of basis](/page/Change%20Of%20Basis) replaces the matrix by a congruent matrix, not by a similar matrix. To compare two matrix formulas for the same geometric form, we need a relation that records how quadratic expressions transform under coordinate substitution. [definition: Congruent Matrices] Let $F$ be a field. Two matrices $A, B \in F^{n \times n}$ are congruent if there exists an invertible matrix $P \in F^{n \times n}$ such that \begin{align*} B = P^\top A P. \end{align*} [/definition] Congruence is the matrix language for changing coordinates in a quadratic form. The next statement records the exact transformation law, and it is the reason congruence rather than similarity is the right [equivalence relation](/page/Equivalence%20Relation) here. [quotetheorem:9909] Thus changing coordinates preserves the underlying quadratic form while changing its representing matrix by congruence. This is why invariants of quadratic forms must be stable under $A \mapsto P^\top A P$, not merely under similarity. Cross terms are where the matrix representation earns its keep. A polynomial such as $2x_1x_2$ has no square terms, but it is still quadratic and comes from a symmetric matrix. [example: A Cross-Term Quadratic Form] Define $Q: \mathbb{R}^2 \to \mathbb{R}$ by \begin{align*} Q(x_1,x_2)=2x_1x_2. \end{align*} Let \begin{align*} A=\begin{pmatrix}0&1\cr 1&0\end{pmatrix}. \end{align*} For $x=(x_1,x_2)^\top$, we compute \begin{align*} Ax=\begin{pmatrix}0&1\cr 1&0\end{pmatrix}\begin{pmatrix}x_1\cr x_2\end{pmatrix}=\begin{pmatrix}x_2\cr x_1\end{pmatrix}. \end{align*} Therefore \begin{align*} x^\top Ax=\begin{pmatrix}x_1&x_2\end{pmatrix}\begin{pmatrix}x_2\cr x_1\end{pmatrix}=x_1x_2+x_2x_1=2x_1x_2=Q(x_1,x_2). \end{align*} Thus $Q(x)=x^\top A x$ with $A_{12}=A_{21}=1$ and $A_{11}=A_{22}=0$. Now set \begin{align*} u=x_1+x_2 \quad \text{and} \quad v=x_1-x_2. \end{align*} Adding the two equations gives \begin{align*} u+v=(x_1+x_2)+(x_1-x_2)=2x_1, \end{align*} so $x_1=(u+v)/2$. Subtracting the second equation from the first gives \begin{align*} u-v=(x_1+x_2)-(x_1-x_2)=2x_2, \end{align*} so $x_2=(u-v)/2$. Hence \begin{align*} Q(x_1,x_2)=2x_1x_2=2\left(\frac{u+v}{2}\right)\left(\frac{u-v}{2}\right)=\frac{(u+v)(u-v)}{2}. \end{align*} Expanding the numerator, \begin{align*} (u+v)(u-v)=u^2-uv+vu-v^2=u^2-v^2, \end{align*} because real multiplication is commutative, so $uv=vu$. Therefore \begin{align*} Q(x_1,x_2)=\frac{1}{2}u^2-\frac{1}{2}v^2. \end{align*} The original formula has only a cross term, but the coordinates $u=x_1+x_2$ and $v=x_1-x_2$ reveal one positive square direction and one negative square direction. [/example] ## Equivalent Characterisations The diagonal expression $q(v)=B(v,v)$ loses the off-diagonal values of $B$, but in many settings those values can be recovered from $q$. The recovery formula is called polarization, and it explains why quadratic forms are often treated as another language for symmetric bilinear forms. [quotetheorem:426] This identity is the bridge between the polynomial and bilinear viewpoints. It also shows exactly why characteristic $2$ is special: the coefficient $1/2$ may not exist. Thus, over fields where $2$ is invertible, the diagonal function $q(v)=B(v,v)$ and the symmetric bilinear form $B$ determine one another. That settles the coordinate-free equivalence, but it does not yet give the most convenient coordinate shape for calculation. Cross terms such as $x_ix_j$ mix coordinate directions: changing one coordinate can affect the contribution of another, so signs, zeros, and independent pieces are no longer visible term by term. The obstruction is therefore not in the definition of a quadratic form, but in the chosen basis. We want a basis in which the polynomial separates into one square contribution from each coordinate direction, with no mixed terms left. The following definition records exactly that coordinate situation. [definition: Diagonal Quadratic Form] Let $F$ be a field and let $V$ be a finite-dimensional vector space over $F$ with ordered basis $\mathcal B = (e_1, \ldots, e_n)$. A quadratic form $q: V \to F$ is diagonal in the basis $\mathcal B$ if there exist $a_1, \ldots, a_n \in F$ such that \begin{align*} q(v) = \sum_{i=1}^n a_i c_i^2 \end{align*} whenever $[v]_{\mathcal B} = (c_1, \ldots, c_n)^\top$. [/definition] Diagonalization is useful because it turns geometric questions into arithmetic ones. The natural theorem to ask for is therefore an existence theorem: under what hypotheses can every finite-dimensional quadratic form be put into diagonal form? [quotetheorem:427] Diagonalization does not say that all diagonal forms are equivalent. It reduces classification to understanding which diagonal coefficients can be changed by rescaling basis vectors and by further congruences. ## Standard Examples The Euclidean norm squared is the model positive quadratic form. It is the algebraic core behind length in Euclidean geometry and the simplest example of a quadratic form coming from an [inner product](/page/Inner%20Product). [example: Euclidean Norm Squared] Let $Q: \mathbb{R}^n \to \mathbb{R}$ be defined by \begin{align*} Q(x)=|x|^2=\sum_{i=1}^n x_i^2. \end{align*} Since $I_nx=x$, matrix multiplication gives \begin{align*} x^\top I_nx=x^\top x=\sum_{i=1}^n x_i x_i=\sum_{i=1}^n x_i^2=Q(x). \end{align*} Thus $Q(x)=x^\top I_nx$ with $I_n$ symmetric, so $Q$ is a real quadratic form. For its associated bilinear form, take $x,y \in \mathbb{R}^n$. Expanding each coordinate square, \begin{align*} Q(x+y)=\sum_{i=1}^n (x_i+y_i)^2=\sum_{i=1}^n (x_i^2+2x_iy_i+y_i^2). \end{align*} Therefore \begin{align*} Q(x+y)-Q(x)-Q(y)=\sum_{i=1}^n (x_i^2+2x_iy_i+y_i^2)-\sum_{i=1}^n x_i^2-\sum_{i=1}^n y_i^2=2\sum_{i=1}^n x_iy_i. \end{align*} Using the polarization formula from the definition of the associated bilinear form, \begin{align*} B_Q(x,y)=\frac{1}{2}\bigl(Q(x+y)-Q(x)-Q(y)\bigr)=\sum_{i=1}^n x_iy_i=x \cdot y. \end{align*} Finally, if $x \ne 0$, then some coordinate $x_k$ is nonzero, so $x_k^2>0$ and every $x_i^2 \ge 0$. Hence \begin{align*} Q(x)=\sum_{i=1}^n x_i^2 \ge x_k^2>0. \end{align*} This is the model positive quadratic form: its values are exactly squared Euclidean lengths. [/example] The next example shows why quadratic forms are not just norms in disguise. A quadratic form can take positive, negative, and zero values on nonzero vectors, even when its formula is nonzero. [example: Hyperbolic Plane] Let $q: \mathbb{R}^2 \to \mathbb{R}$ be defined by \begin{align*} q(x_1,x_2)=x_1^2-x_2^2. \end{align*} Set \begin{align*} A=\begin{pmatrix}1&0\cr 0&-1\end{pmatrix}. \end{align*} For $x=(x_1,x_2)^\top$, matrix multiplication gives \begin{align*} Ax=\begin{pmatrix}1&0\cr 0&-1\end{pmatrix}\begin{pmatrix}x_1\cr x_2\end{pmatrix}=\begin{pmatrix}x_1\cr -x_2\end{pmatrix}. \end{align*} Therefore \begin{align*} x^\top Ax=\begin{pmatrix}x_1&x_2\end{pmatrix}\begin{pmatrix}x_1\cr -x_2\end{pmatrix}=x_1x_1+x_2(-x_2)=x_1^2-x_2^2=q(x_1,x_2). \end{align*} Thus $q(x)=x^\top A x$, and $A$ is symmetric because its off-diagonal entries are both $0$. The sign behavior is visible on three simple vectors. For $(1,0)$, \begin{align*} q(1,0)=1^2-0^2=1-0=1. \end{align*} For $(0,1)$, \begin{align*} q(0,1)=0^2-1^2=0-1=-1. \end{align*} For the nonzero vector $(1,1)$, \begin{align*} q(1,1)=1^2-1^2=1-1=0. \end{align*} So this quadratic form takes positive, negative, and zero values on nonzero inputs; it is indefinite rather than positive definite. [/example] ## Properties The most basic qualitative property of a real quadratic form is whether it is positive, negative, or mixed in sign. This matters in optimization because the second-order behavior of a smooth function near a critical point is governed by a quadratic form, namely the Hessian quadratic form. [definition: Positive Definite Quadratic Form] A real quadratic form $Q: \mathbb{R}^n \to \mathbb{R}$ is positive definite if \begin{align*} Q(x)>0 \end{align*} for every nonzero $x \in \mathbb{R}^n$. [/definition] Positive definiteness is stronger than nonnegativity. Many important quadratic forms are allowed to vanish in some nonzero directions, especially when a quantity measures only part of a vector. This motivates a separate semidefinite notion. [definition: Positive Semidefinite Quadratic Form] A real quadratic form $Q: \mathbb{R}^n \to \mathbb{R}$ is positive semidefinite if \begin{align*} Q(x) \ge 0 \end{align*} for every $x \in \mathbb{R}^n$. [/definition] Semidefinite forms arise naturally when a squared quantity has directions of degeneracy. To distinguish these from forms that genuinely take both signs, we need a term for the mixed-sign case. [definition: Indefinite Quadratic Form] A real quadratic form $Q: \mathbb{R}^n \to \mathbb{R}$ is indefinite if there exist vectors $u,v \in \mathbb{R}^n$ such that \begin{align*} Q(u)>0 \quad \text{and} \quad Q(v)<0. \end{align*} [/definition] Indefinite forms appear in saddle points, Lorentzian geometry, and constrained variational problems. Once these sign types have been named, the computational question is how to read them from a matrix without testing every vector. [quotetheorem:3281] This criterion explains why symmetric matrices are the correct matrices for real quadratic forms: their eigenvalues are real and their eigenspaces give orthogonal axes for the form. Positivity alone is too coarse for many geometric questions. A saddle with one downward direction behaves differently from a saddle with many downward directions, and a zero direction signals degeneracy rather than genuine curvature. The next invariant records that finer shape by counting the positive, negative, and zero axes after diagonalization, so it distinguishes forms that the broad labels "positive semidefinite" and "indefinite" cannot separate. [definition: Signature of a Real Quadratic Form] Let $Q: \mathbb{R}^n \to \mathbb{R}$ be a real quadratic form. A signature of $Q$ is a triple $(p,m,z)$ where, in a diagonal expression for $Q$, the number of positive diagonal coefficients is $p$, the number of negative diagonal coefficients is $m$, and the number of zero diagonal coefficients is $z$. [/definition] The signature is intended to describe the geometry of the form, so it should not depend on the particular diagonalization chosen. A possible obstruction is that different changes of basis might produce diagonal forms with different numbers of positive and negative entries. The key invariance question is whether those counts are forced by the original form itself. [quotetheorem:429] Sylvester's law says that real quadratic forms have a stable geometry under coordinate changes. Rotations, scalings, and shears can simplify the formula, but they cannot turn a saddle form into a positive definite one. A different issue is whether the form has invisible directions for its associated bilinear pairing. This motivates the radical, which measures the vectors that pair to zero with every vector and therefore cannot be detected by the bilinear structure. [definition: Radical of a Quadratic Form] Let $F$ be a field with $\operatorname{char}(F) \ne 2$, let $V$ be a vector space over $F$, and let $q: V \to F$ be a quadratic form with associated symmetric bilinear form $B: V \times V \to F$. The radical of $q$ is \begin{align*} \operatorname{rad}(q)=\{v \in V : B(v,w)=0 \text{ for every } w \in V\}. \end{align*} [/definition] The radical is the subspace of vectors that pair to zero with everything. Many classification and duality arguments require ruling out such invisible directions, which leads to the nondegenerate case. [definition: Nondegenerate Quadratic Form] Let $F$ be a field with $\operatorname{char}(F) \ne 2$, let $V$ be a vector space over $F$, and let $q: V \to F$ be a quadratic form. The form $q$ is nondegenerate if \begin{align*} \operatorname{rad}(q)=\{0\}. \end{align*} [/definition] The definition of nondegeneracy is intrinsic, but computations usually begin with a matrix representative. To use the definition directly, one would have to solve $B(v,w)=0$ for every vector $w$, which is an infinite-looking condition. A matrix turns this into the question of whether the associated [linear map](/page/Linear%20Map) from $V$ to its dual has a kernel, and the determinant is the finite test for that failure. [quotetheorem:7661] The determinant condition is basis-independent because changing basis replaces $A$ by $P^\top A P$, whose determinant is $(\det P)^2\det A$. ## Relationship to Other Concepts Quadratic forms sit immediately beside bilinear forms. In characteristic not equal to $2$, the polarization identity makes this relationship reversible for symmetric bilinear forms. In characteristic $2$, quadratic forms have extra behavior, and the associated polar form can be alternating even when the quadratic form is not zero. They also provide the natural language for second derivatives. If $f: U \subset \mathbb{R}^n \to \mathbb{R}$ is twice continuously differentiable and $x_0 \in U$ is a critical point, then the Hessian matrix $Hf_{x_0}$ defines the quadratic form \begin{align*} h \mapsto h^\top Hf_{x_0}h. \end{align*} The second-order term in Taylor's formula is \begin{align*} \frac{1}{2}h^\top Hf_{x_0}h, \end{align*} and the factor $1/2$ does not change its sign. When this form is positive definite, negative definite, or indefinite, its sign gives the usual second-derivative test for a local minimum, local maximum, or saddle point; in semidefinite cases, higher-order terms may decide the behavior. This connects quadratic forms to [derivatives](/page/Derivative), Taylor polynomial, and optimization. In geometry, positive definite quadratic forms on each tangent space, through their associated symmetric bilinear forms, are the local data of a Riemannian metric. Indefinite nondegenerate forms lead instead toward pseudo-Riemannian geometry, where nonzero vectors may have zero length. In number theory, integral quadratic forms ask representation questions: for which integers $m$ does there exist $x \in \mathbb{Z}^n$ such that $q(x)=m$? This connects the theory to congruences, lattices, and arithmetic invariants. [remark: Quadratic Forms Versus Quadratic Polynomials] A quadratic form is homogeneous of degree $2$. A general quadratic polynomial may also contain linear and constant terms, such as $x^2+3x+1$. Homogeneity is what makes the identity $q(av)=a^2q(v)$ possible. [/remark] This distinction becomes especially delicate when the characteristic changes. The next warning records the main place where formulas from the characteristic-not-$2$ theory stop behaving as recovery formulas. [remark: Characteristic Two Warning] Over a field of characteristic $2$, the formula $q(u+v)-q(u)-q(v)$ can still define a bilinear polar form, but it need not recover $q$. Many classification statements must be modified in that setting. [/remark] ## Beyond and Connections Quadratic forms sit at a crossroads between linear algebra, geometry, arithmetic, and topology. Over the real numbers, diagonalization and signature turn a form into a geometric object: positive definite forms give Euclidean lengths, indefinite forms give hyperbolic and Lorentzian geometries, and degenerate forms record directions that the quadratic expression cannot detect. Over fields other than $\mathbb{R}$, the same normal-form questions become subtler. Square classes, discriminants, and related invariants measure what diagonal changes of variables cannot remove. This is why quadratic forms naturally lead to arithmetic topics such as forms over $\mathbb{Q}$, $p$-adic fields, and finite fields, where local solvability and equivalence can differ sharply from the real case. The associated bilinear form connects the subject to symmetric matrices and [self-adjoint operators](/page/Self-Adjoint%20Operators): after a basis is chosen, a quadratic form is represented by a symmetric matrix, and change of basis acts by congruence rather than similarity. Thus the invariants of quadratic forms are not eigenvalue invariants alone; they are invariants of the underlying bilinear geometry. Quadratic forms also provide local models in analysis and geometry. The Hessian of a smooth function at a critical point is a quadratic form, so definiteness and signature control local extrema and saddle behavior. In topology and differential geometry, intersection forms and metric tensors are higher-level versions of the same idea: a bilinear pairing encodes geometry, while its quadratic expression records lengths, signs, and curvature-like information. ## References Sheldon Axler, *Linear Algebra Done Right* (2015). Jean-Pierre Serre, *A Course in Arithmetic* (1973). John Milnor and Dale Husemoller, *Symmetric Bilinear Forms* (1973). [Symmetric Matrix](/page/Symmetric%20Matrix). [Bilinear Form](/page/Bilinear%20Form). [Inner Product Space](/page/Inner%20Product%20Space).