Bilinear and Quadratic Forms

A bilinear form is the two-slot generalization of the dot product: a scalar-valued machine $B:V\times V\to\mathbb{F}$ that is linear in each argument separately, so once a basis is fixed it becomes a matrix, $B(x,y)=x^\mathsf{T}Ay$ . Close both slots onto the same vector and you get a quadratic form $Q(x)=B(x,x)=x^\mathsf{T}Ax$ — a homogeneous degree-two polynomial in the coordinates, the object that appears every time physics writes down an energy, a metric, or a second-order expansion of a potential. These forms sit one layer beyond linear operators, and the difference is not cosmetic: an operator returns a vector, so it transforms by similarity $A\mapsto P^{-1}AP$ ; a form returns a number, so it transforms by congruence $A\mapsto P^\mathsf{T}AP$ . Getting that one transpose right, or wrong, is the whole game. The deepest fact about real symmetric forms is that congruence has almost nothing to remember. Sylvester's law of inertia says that no matter how you diagonalize $Q$ — by rotating, shearing, rescaling — the numbers of positive, negative, and zero diagonal entries never change. Those three integers $(n_+,n_-,n_0)$ , the signature, are the complete congruence invariant, and they are what make "positive-definite", "Lorentzian", and "indefinite" meaningful adjectives rather than accidents of a chosen basis. Two diagonalization routes coexist: Lagrange's completion of squares, fast and metric-blind, delivering $\pm1$ s; and the principal-axis theorem, an orthogonal congruence built from eigenvectors, where congruence and similarity coincide and the diagonal entries are the genuine eigenvalues that set the semi-axes of a quadric. This article grounds the algebra, proves the law, and shows why one invariant decides whether an equilibrium is stable, a conic is an ellipse, or a spacetime is causal.

Key takeaways

A bilinear form is linear in each slot, $B(x,y)=x^\mathsf{T}Ay$ ; its diagonal restriction $Q(x)=x^\mathsf{T}Ax$ is a quadratic form, and only the symmetric part of $A$ survives.
Forms transform by congruence $A\mapsto P^\mathsf{T}AP$ , not similarity $P^{-1}AP$ — the two agree only for orthogonal $P$ .
Sylvester's law of inertia: the counts $(n_+,n_-,n_0)$ of positive, negative, and zero squares are congruence-invariant; the signature classifies a real form completely.
Positive-definiteness is equivalent to all eigenvalues positive, all leading principal minors positive (Sylvester's criterion), and a Cholesky factorization $A=LL^\mathsf{T}$ .
The principal-axis theorem diagonalizes $Q$ by an orthogonal congruence from the eigenvectors of $A$ , giving $Q=\sum_i\lambda_i y_i^2$ while preserving lengths and angles.
Signature is the physics: it fixes stability of equilibria, the shape of conics and quadrics, and the Riemannian-versus-Lorentzian character of a metric.
Key idea: A quadratic form is a symmetric matrix viewed through congruence rather than similarity; its only real invariants are the eigenvalue signs (the signature), and diagonalizing it means either completing squares (Lagrange) or rotating to principal axes (spectral theorem).

The core insight

A quadratic form is a symmetric matrix viewed through congruence rather than similarity; its only real invariants are the eigenvalue signs — the signature $(n_+,n_-,n_0)$ — and diagonalizing it means either completing squares (Lagrange) or rotating to principal axes (the spectral theorem).

Where this lives today

Quadratic forms are the daily arithmetic of computational physics. Every second-order optimization method — Newton's method, trust-region and interior-point solvers behind machine learning and structural design — tests the Hessian's signature at each step to tell a minimum from a saddle, and factors it by Cholesky when positive-definite. Molecular and lattice dynamics diagonalize the mass-weighted Hessian to extract normal-mode frequencies $\omega_i=\sqrt{\lambda_i}$ ; an imaginary frequency — a negative eigenvalue, a defect in the signature — flags a transition state or a soft phonon mode that triggers a phase transition. In numerical relativity the signature $(1,3)$ is monitored as a hard constraint, since a code that drifts to the wrong signature has lost causal structure. Statistics leans on the same object: a covariance matrix is a positive-semidefinite form, and principal component analysis is its principal-axis decomposition. And in convex optimization, semidefinite programming optimizes directly over the cone of positive-semidefinite matrices, making Sylvester's classification the geometry a solver navigates in real time.

The physical picture

Picture a quadratic form as a landscape over the vector space: at each point $x$ the height is $Q(x)=x^\mathsf{T}Ax$ . Because $Q$ is homogeneous of degree two, doubling $x$ quadruples the height, so the entire shape is encoded in one level set $Q(x)=1$ : a conic in the plane, a quadric in space. It is an ellipse if the bowl opens upward in every direction, a hyperbola if it climbs some ways and falls others, a straight-walled trough if it is flat along a direction where $Q$ vanishes. The eigenvectors of $A$ point along the principal axes and the eigenvalues are the reciprocal squared semi-axes, signs included. Sylvester's law says you may shear and rescale the picture however you like but can never turn a bowl into a saddle: the tally of upward versus downward directions is fixed.

Figure

The level set Q equal to 1 sorted by signature: a positive-definite form (2, 0) is an ellipse, an indefinite (1, 1) a hyperbola whose asymptotes are the light cone Q equal to 0, a degenerate (1, 0, 1) a straight trough. For the ellipse λ = 3, 7 so det A = λ₁λ₂ = 21 and both signs agree

The reason a form obeys $P^\mathsf{T}AP$ while an operator obeys $P^{-1}AP$ is pure index bookkeeping. A form carries two lower (covariant) indices $B_{ij}$ , so a basis change contracts $P$ onto both slots and $P$ appears twice, once transposed; an operator carries one upper and one lower index $A^i{}_j$ , so $P$ appears once with its inverse. The two collapse into one only for orthogonal $P$ — exactly what the principal-axis theorem exploits: rotate to the eigenvectors of $A$ , the cross terms vanish, the diagonal entries become the true eigenvalues, and the metric is untouched. The second figure makes that rotation concrete.

Figure

Principal-axis rotation of Q = 5x² + 4xy + 5y² = 1. The tilted ellipse carries a 4xy cross term; rotating 45° onto the eigenvectors u = (1,1)/√2 and v = (1,−1)/√2 removes it, leaving Q = 7u² + 3v². The eigenvalues λ = 7, 3 set the semi-axes 1/√7 = 0.378 and 1/√3 = 0.577

The core equations

A bilinear form is defined by one property: linearity in each argument with the other held fixed. Everything algebraic follows from squeezing that into a basis.

B(\alpha x_1+\beta x_2,\,y)=\alpha B(x_1,y)+\beta B(x_2,y),\qquad B(x,\alpha y_1+\beta y_2)=\alpha B(x,y_1)+\beta B(x,y_2) \qquad (1)

where: $B$ — the bilinear form, a map $V\times V\to\mathbb{F}$ ; $x,y$ — vectors in $V$ ; $\alpha,\beta$ — scalars in the field $\mathbb{F}$ . Reading: fixing one slot and varying the other reproduces ordinary linearity, so $B$ is determined by its action on a basis. Fixing a basis $\{e_i\}$ , bilinearity forces $B$ into a matrix sandwich. The entries $A_{ij}=B(e_i,e_j)$ form the Gram matrix, and every evaluation is a double sum over coordinates.

B(x,y)=x^\mathsf{T}Ay=\sum_{i,j}A_{ij}\,x_i y_j,\qquad A_{ij}=B(e_i,e_j) \qquad (2)

where: $A$ — Gram matrix of $B$ in the basis $\{e_i\}$ ; $x,y$ — coordinate column vectors of the two arguments. Reading: the two-slot machine and the matrix carry the same information once a basis is chosen. The quadratic form is the diagonal restriction $Q(x)=B(x,x)$ . The antisymmetric part of $A$ contributes nothing — $x^\mathsf{T}Kx$ is a scalar equal to its own transpose $-x^\mathsf{T}Kx$ , so it vanishes — so one always symmetrizes, $A\to\tfrac12(A+A^\mathsf{T})$ , and takes $A=A^\mathsf{T}$ from the outset.

Q(x)=x^\mathsf{T}Ax=\sum_{i,j}A_{ij}\,x_i x_j,\qquad A=A^\mathsf{T} \qquad (3)

where: $Q$ — the quadratic form $Q(x)=B(x,x)$ ; $A$ — the symmetric matrix $\tfrac12(A+A^\mathsf{T})$ , the only part that survives. Reading: a quadratic form and a symmetric matrix are two names for one object. Under $x=Py'$ with $P$ invertible, both arguments carry a factor of $P$ , so the matrix transforms by congruence — note the transpose, not an inverse. This single line is the source of every subtlety that follows.

x=Py'\ \Longrightarrow\ Q(x)=y'^\mathsf{T}(P^\mathsf{T}AP)\,y',\qquad A'=P^\mathsf{T}AP \qquad (4)

where: $P$ — invertible change-of-basis matrix; $A'$ — the matrix of the same form in the new coordinates. Reading: for an operator one would instead get $A'=P^{-1}AP$ (similarity); the two coincide only when $P^\mathsf{T}=P^{-1}$ , i.e. $P$ orthogonal. Sylvester's law of inertia guarantees any real symmetric form can be congruence-diagonalized into a canonical string of $+1$ , $-1$ , and $0$ — and, crucially, that the three counts are unique.

P^\mathsf{T}AP=\operatorname{diag}(\underbrace{1,\dots,1}_{n_+},\,\underbrace{-1,\dots,-1}_{n_-},\,\underbrace{0,\dots,0}_{n_0}) \qquad (5)

where: $n_+,n_-,n_0$ — the numbers of positive, negative, and zero entries, invariant under any congruence; $(n_+,n_-)$ — the signature, with $n_++n_-+n_0=n$ . Reading: this canonical form is unique even though the diagonalizing $P$ is wildly non-unique. When the congruence is chosen orthogonal — built from the orthonormal eigenvectors of $A$ — congruence and similarity fuse and the diagonal entries become the actual eigenvalues, the principal-axis theorem.

A=O\Lambda O^\mathsf{T},\qquad O^\mathsf{T}AO=O^{-1}AO=\Lambda=\operatorname{diag}(\lambda_1,\dots,\lambda_n),\qquad O^\mathsf{T}O=I \qquad (6)

where: $O$ — orthogonal matrix whose columns are unit eigenvectors of $A$ ; $\lambda_i$ — the real eigenvalues, the principal-axis coefficients. Reading: under $x=Oy$ the form becomes $Q=\sum_i\lambda_i y_i^2$ with the metric preserved. Definiteness has a determinant test needing no eigenvalues: a symmetric $A$ is positive-definite exactly when every leading principal minor is positive (Sylvester's criterion), equivalent to the existence of a Cholesky factorization.

A\succ 0\ \Longleftrightarrow\ \det(A_k)>0\ \ (k=1,\dots,n)\ \Longleftrightarrow\ A=LL^\mathsf{T},\ \ L_{ii}>0 \qquad (7)

where: $A_k$ — the top-left $k\times k$ leading principal submatrix; $L$ — the unique lower-triangular Cholesky factor with positive diagonal. Reading: $n$ nested positive determinants certify definiteness cheaply, and $A=LL^\mathsf{T}$ is the constructive witness. Finally, the determinant ties to the signature: nonzero exactly when the form is nondegenerate, its sign counting the negative eigenvalues modulo two.

\Delta=\det A=\prod_i\lambda_i,\qquad \operatorname{sign}(\Delta)=(-1)^{n_-}\quad(n_0=0) \qquad (8)

where: $\Delta$ — the discriminant of the form; $n_-$ — the number of negative eigenvalues. Reading: a single determinant already reveals whether an even or odd number of directions "point downward".

Derivation ASylvester's law of inertia

The goal is to prove that the signature is a genuine invariant: if a real quadratic form is reduced to a sum and difference of squares in two entirely different ways, the counts of positive and of negative squares must agree, even though the two sets of diagonalizing coordinates need share nothing. This is what licenses us to speak of the signature and to call definiteness coordinate-free; the argument is a clean dimension count that never touches the diagonalizing transformation, only the sign of $Q$ on well-chosen subspaces.

Step 1. Suppose two invertible changes of coordinate reduce the same form to canonical squares, $Q=\sum_{i=1}^{p}u_i^2-\sum_{i=p+1}^{p+r}u_i^2=\sum_{i=1}^{p'}w_i^2-\sum_{i=p'+1}^{p'+r'}w_i^2$ , the remaining coordinates lying in the radical. The nonzero counts satisfy $p+r=p'+r'=n-n_0$ since rank is basis-independent. Why: congruence preserves rank, so only the split between positive and negative squares is in question.

Step 2. Let $U_+$ be the subspace where the first coordinates have $u_{p+1}=\dots=u_n=0$ ; there $Q(x)=\sum_{i=1}^p u_i^2>0$ for nonzero $x$ , and $\dim U_+=p$ . Let $W_{\le}$ be the subspace where the second set has $w_1=\dots=w_{p'}=0$ ; there $Q(x)=-\sum w_i^2\le 0$ , and $\dim W_{\le}=n-p'$ . Why: each diagonalization hands us a subspace of definite sign whose dimension is one of the counts we compare.

Step 3. These subspaces meet only at the origin: any $x$ in $U_+\cap W_{\le}$ would satisfy $Q(x)>0$ and $Q(x)\le 0$ at once, impossible unless $x=0$ , so the sum $U_+ + W_{\le}$ is direct. Why: a number cannot be both strictly positive and nonpositive, so the two subspaces are transverse.

Step 4. A direct sum cannot overflow the space, so $\dim U_+ + \dim W_{\le}=p+(n-p')\le n$ , giving $p\le p'$ . Swapping the two roles gives $p'\le p$ , hence $p=p'$ ; the same argument on $-Q$ fixes the negative count, and $n_0=n-n_+-n_-$ follows. So $(n_+,n_-,n_0)$ is an invariant of the congruence class. Why: two opposite inequalities force equality, and the whole conclusion rests on dimension counting, not on any explicit transformation. *Why the law bites immediately in $\mathbb{R}^2$ : a nondegenerate form with $\det A<0$ must have signature $(1,1)$ , since $\lambda_1\lambda_2=\det A<0$ forces the two eigenvalues to have opposite signs regardless of the diagonalizing basis.*

Go deeper · Derivation B · 1 minLagrange reduction by completing the square

Derivation BLagrange reduction by completing the square

The goal is an explicit, eigenvalue-free algorithm that reduces any quadratic form to a weighted sum of squares by a triangular (hence invertible) change of variables, exposing the signature directly, and to see on a worked case why the resulting coefficients are emphatically not the eigenvalues — they merely share the eigenvalues' signs, all Sylvester's law promises. We take $Q(x_1,x_2)=x_1^2+4x_1x_2+x_2^2$ and complete the square one variable at a time.

Step 1. Collect every term containing $x_1$ and complete the square: $Q=(x_1^2+4x_1x_2)+x_2^2=(x_1^2+4x_1x_2+4x_2^2)-4x_2^2+x_2^2$ , adding and subtracting $4x_2^2$ . Why: this folds all of $x_1$ 's cross-coupling into one perfect square, isolating a decoupled variable.

Step 2. The bracket is a perfect square, so $Q=(x_1+2x_2)^2-3x_2^2$ . Define $u_1=x_1+2x_2$ , $u_2=x_2$ : upper-triangular with unit diagonal, hence invertible with determinant $1$ . Why: a triangular substitution is automatically an admissible congruence that loses no information.

Step 3. In the new coordinates $Q=u_1^2-3u_2^2$ : one positive and one negative coefficient, so the signature is $(1,1)$ — indefinite. Rescaling $u_2\to\sqrt3\,u_2$ gives the Sylvester normal form $u_1^2-u_2^2$ . Why: only the coefficient signs are congruence-invariant, so rescaling to $\pm1$ loses nothing classification cares about.

Step 4. Cross-check against $A=\begin{pmatrix}1&2\\2&1\end{pmatrix}$ : $\det A=-3$ , $\operatorname{tr}A=2$ give $\lambda=3,-1$ , the same signs as the Lagrange coefficients $1,-3$ but different magnitudes. Why: this is the tangible content of "congruence preserves signs, not values" — completed-square weights and eigenvalues need never coincide. *Why the signs agree: expanding $(x_1+2x_2)^2-3x_2^2$ recovers $x_1^2+4x_1x_2+x_2^2$ exactly, and the spectrum $\{3,-1\}$ of $A$ carries the same sign pattern $(+,-)$ as the Lagrange coefficients — the tangible content of congruence preserving signs but not magnitudes.*

Go deeper · Derivation C · 1 minPrincipal axes from the spectral theorem

Derivation CPrincipal axes from the spectral theorem

The goal is to show that for a real symmetric matrix the diagonalizing congruence may be chosen orthogonal, so the same rotation is at once a similarity and a congruence, the diagonal entries are the true eigenvalues, and the Euclidean metric is preserved — the theorem that turns abstract classification into concrete geometry, giving the principal axes and semi-axes of any conic or quadric and the principal moments of any inertia tensor.

Step 1. Eigenvalues are real. From $Av=\lambda v$ , form $\lambda\,v^\dagger v=v^\dagger Av=(v^\dagger Av)^\dagger=\bar\lambda\,v^\dagger v$ , using $A=A^\mathsf{T}=A^\dagger$ ; since $v^\dagger v>0$ , $\lambda=\bar\lambda$ is real. Why: real eigenvalues are what make the signature — a count of real signs — well posed.

Step 2. Eigenvectors for distinct eigenvalues are orthogonal: $\lambda_i\,v_i^\mathsf{T}v_j=(Av_i)^\mathsf{T}v_j=v_i^\mathsf{T}Av_j=\lambda_j\,v_i^\mathsf{T}v_j$ , so $(\lambda_i-\lambda_j)\,v_i^\mathsf{T}v_j=0$ and $v_i\perp v_j$ when $\lambda_i\ne\lambda_j$ ; repeated eigenvalues are handled by Gram–Schmidt in each eigenspace. Why: orthogonal eigenvectors let us build an orthogonal, metric-preserving diagonalizer.

Step 3. Assemble the columns into $O=[\,v_1\ \cdots\ v_n\,]$ with the $v_i$ orthonormal, so $O^\mathsf{T}O=I$ and $O^{-1}=O^\mathsf{T}$ . The eigenvalue equations bundle into $AO=O\Lambda$ . Why: orthonormal columns make $O$ orthogonal, the one case where inverse and transpose agree.

Step 4. Left-multiply by $O^\mathsf{T}=O^{-1}$ to get $O^\mathsf{T}AO=\Lambda$ — one equation that is at once a similarity and a congruence. Under $x=Oy$ the form becomes $Q=\sum_i\lambda_i y_i^2$ , and since $\lVert Oy\rVert=\lVert y\rVert$ lengths and angles are untouched. Why: this is why the eigenvalues, not arbitrary Lagrange weights, set the true semi-axes — the rotation preserves the metric that measures them. *Why the rotation works: for $A=\begin{pmatrix}5&2\\2&5\end{pmatrix}$ , $O$ is the $45^\circ$ rotation, $\Lambda=\operatorname{diag}(7,3)$ , and $O^\mathsf{T}AO=\Lambda$ gives $Q=7u^2+3v^2$ with the unit circle preserved — confirming that the orthogonal congruence is simultaneously a similarity and a metric-preserving rotation.*

Limiting cases

The signature $(n_+,n_-,n_0)$ partitions all real symmetric forms into qualitatively distinct regimes; each row below is a physically named class.

Case	Result	Meaning
Positive-definite	$n_+=n,\ \ Q(x)>0\ \forall x\ne 0$	All eigenvalues positive; a bounded ellipsoid level set and a genuine inner product. Energy and Riemannian metric tensors, stable minima.
Negative-definite	$n_-=n,\ \ Q(x)<0\ \forall x\ne 0$	Mirror image; $-Q$ is an inner product. A strict local maximum of a potential.
Indefinite (Lorentzian)	$\operatorname{sig}=(1,n-1)$ or $(n-1,1)$	Mixed signs; a hyperboloid level set and a light cone $Q=0$ . Spacetime metric $\operatorname{diag}(+,-,-,-)$ ; saddle equilibria.
Positive-semidefinite (degenerate)	$n_0>0,\ n_-=0,\ Q(x)\ge 0$	A nontrivial radical where $Q$ vanishes; the form descends to a definite one on the quotient. Constrained and gauge systems.

Common trapTransforming a form as if it were an operator

The most common error is to change a form's matrix by similarity, $P^{-1}AP$ , instead of by congruence, $P^\mathsf{T}AP$ . Under $x=Py$ a form's matrix becomes $P^\mathsf{T}AP$ because both arguments carry the transformation — full stop. The two agree only for orthogonal $P$ , which is why the principal-axis route is special and why Lagrange reduction (non-orthogonal $P$ ) yields $\pm1$ coefficients that are not the eigenvalues. A quieter version of the same mistake: writing $Q(x)=x^\mathsf{T}Ax$ with a non-symmetric $A$ and trusting eigenvalue formulas. Always symmetrize first — the antisymmetric part contributes zero to $Q$ but silently corrupts every determinant, eigenvalue, and definiteness test that assumes $A=A^\mathsf{T}$ .

Worked examples

1 · Classify and diagonalize a conic (fully worked)

The curve $Q(x,y)=5x^2+4xy+5y^2=1$ is a conic. Decide its type, find its principal axes and semi-axes, and remove the cross term.

Step 1. Build the symmetric matrix by splitting the cross term evenly: $A=\begin{pmatrix}5&2\\2&5\end{pmatrix}$ , the off-diagonal $2$ being half the coefficient $4$ of $xy$ — the standard symmetrization.

Step 2. Test definiteness with Sylvester's criterion first: leading minors $\det A_1=5>0$ and $\det A_2=\det A=25-4=21>0$ . Both positive, so $A\succ 0$ — the form is positive-definite and the conic is a bounded ellipse.

Step 3. Eigenvalues from $\operatorname{tr}A=10$ , $\det A=21$ : the characteristic equation $\lambda^2-10\lambda+21=(\lambda-7)(\lambda-3)=0$ gives $\lambda=7,3$ , both positive — confirming Step 2 independently.

Step 4. Eigenvectors: $\lambda=7$ gives $u=(1,1)/\sqrt2$ , $\lambda=3$ gives $v=(1,-1)/\sqrt2$ . They are orthogonal, as the spectral theorem demands, and $O=\tfrac{1}{\sqrt2}\begin{pmatrix}1&1\\1&-1\end{pmatrix}$ is the $45^\circ$ rotation onto principal axes.

Step 5. In principal coordinates $Q=7u^2+3v^2=1$ , so the semi-axes are $a_u=1/\sqrt7=\mathbf{0.378}$ (short, along $u$ ) and $a_v=1/\sqrt3=\mathbf{0.577}$ (long, along $v$ ): the larger eigenvalue $7$ owns the shorter axis. A positive-definite ellipse, principal axes at 45°, semi-axes 0.378 and 0.577, cross-term-free form $Q=7u^2+3v^2$ .

2 · Positive-definiteness by three routes (fully worked)

Is the Hessian $H=\begin{pmatrix}2&-1&0\\-1&2&-1\\0&-1&2\end{pmatrix}$ — the discrete Laplacian for a chain of three coupled masses near equilibrium — positive-definite, so the equilibrium is stable?

Step 1. Sylvester's criterion. The three leading principal minors are $\det H_1=2$ , $\det H_2=4-1=3$ , and $\det H_3=2(4-1)-(-1)(-2)=6-2=4$ . The sequence $2,3,4$ is strictly positive, so $H\succ 0$ — stable.

Step 2. Eigenvalues, for cross-check: this tridiagonal matrix has spectrum $\lambda_k=2-2\cos\frac{k\pi}{4}$ , giving $\lambda=2-\sqrt2,\ 2,\ 2+\sqrt2\approx\mathbf{0.586,\ 2,\ 3.414}$ . All positive — the same verdict, now with the actual normal-mode stiffnesses.

Step 3. Cholesky, the constructive witness: $H=LL^\mathsf{T}$ with $L=\begin{pmatrix}\sqrt2&0&0\\-\tfrac{1}{\sqrt2}&\sqrt{\tfrac32}&0\\0&-\sqrt{\tfrac23}&\sqrt{\tfrac43}\end{pmatrix}$ , whose positive diagonal squares $2,\tfrac32,\tfrac43$ multiply to $\det H=4$ . Real positive pivots re-prove definiteness. All three tests agree: $H\succ 0$ , the equilibrium is a strict minimum, and the mode stiffnesses are $2\mp\sqrt2$ and $2$ .

3 · The Lorentzian metric is congruence-locked

Show that no change of inertial coordinates can turn the spacetime form $Q=c^2t^2-x^2-y^2-z^2$ into a positive-definite one.

Step 1. In coordinates $(ct,x,y,z)$ the matrix is $\eta=\operatorname{diag}(1,-1,-1,-1)$ , signature $(1,3)$ .

Step 2. A Lorentz transformation is by definition a congruence that preserves $\eta$ : $\Lambda^\mathsf{T}\eta\,\Lambda=\eta$ . By Sylvester's law, any congruence — Lorentz or not — leaves $(n_+,n_-)=(1,3)$ fixed. No basis change can make all four signs agree, so the light cone $Q=0$ is an invariant structure and causal order is coordinate-independent.

Physics of the portrait

The classification of quadratic forms is one of the rare corners of physics whose central theorem is named for the mathematician who saw that only the signs survive — and who gave us the words "matrix" and "signature".

Origins · 1 minJames Joseph Sylvester and how this was discovered

James Joseph Sylvester

1814–1897 · the algebra of forms, invariants, and the law of inertia

James Joseph Sylvester was an English mathematician whose fingerprints are on the very vocabulary of this subject: he coined the word "matrix" in 1850, named the "discriminant", and introduced "signature" for the pair of counts that classify a real form. Barred for decades from an English academic career because, as a Jew, he would not subscribe to the Thirty-Nine Articles required at Cambridge, he took his degree at Trinity College Dublin and worked as an actuary and a barrister before mathematics reclaimed him. In 1852 he stated and proved the law of inertia: that when a real quadratic form is reduced to a sum and difference of squares, the numbers of positive and of negative terms are invariants of the form, independent of how the reduction is done — the theorem that makes "positive-definite" and "Lorentzian" mean something. With his lifelong collaborator Arthur Cayley he founded the theory of algebraic invariants, the two often called the "invariant twins". In 1876, at sixty-two, he crossed the Atlantic to become the first professor of mathematics at the new Johns Hopkins University, where he founded the American Journal of Mathematics, effectively seeding graduate mathematics in America, before returning to England in 1883 as Savilian Professor of Geometry at Oxford. His criterion for positive-definiteness through leading principal minors, and the signature that bears his law, are the working tools every physicist reaches for in asking whether an equilibrium is stable or a metric is Riemannian.

History

1801

— Gauss develops the theory of binary quadratic forms in Disquisitiones Arithmeticae

1829

— Cauchy proves real symmetric matrices have real eigenvalues and orthogonal principal axes

1852

— Sylvester states and proves the law of inertia for real quadratic forms

1858

— Cayley's Memoir on the Theory of Matrices gives forms a matrix algebra

1924

— Cholesky's factorization for positive-definite matrices is published posthumously

Going further

Texts. Stone & Goldbart, Mathematics for Physics, linear algebra and quadratic forms · Arfken, Weber & Harris, Mathematical Methods for Physicists, 7th ed., ch. 3 · Hassani, Mathematical Physics, inner-product spaces and spectral decomposition · Byron & Fuller, Mathematics of Classical and Quantum Physics, ch. 3–4 · Nakahara, Geometry, Topology and Physics, §2 (metric signature) · Horn & Johnson, Matrix Analysis, ch. 4 & 7. Key anchor results: the spectrum of $\begin{pmatrix}1&2\\2&1\end{pmatrix}$ is $\lambda=3,-1$ (from $\det=-3$ , $\operatorname{tr}=2$ ), consistent with the completed-square identity $x_1^2+4x_1x_2+x_2^2=(x_1+2x_2)^2-3x_2^2$ ; for $\begin{pmatrix}5&2\\2&5\end{pmatrix}$ the eigenvalues are $7,3$ with semi-axes $1/\sqrt7=0.378$ and $1/\sqrt3=0.577$ ; the discrete-Laplacian leading minors are $2,3,4$ and its spectrum is $2\mp\sqrt2,\,2$ ; the inertia proof rests on the dimension count $p+(n-p')\le n$ .

Prerequisites. Linear Operators and Matrix Representations · Change of Basis and Similarity Transformations · Eigenvalues, Eigenvectors, and the Characteristic Equation · Inner-Product Spaces and Norms · Determinants and Their Properties. Leads to. Diagonalization and the Spectral Decomposition · Adjoint and Hermitian Operators · Orthogonality and the Gram–Schmidt Process.

Ask the tutor: Bring me any symmetric matrix and I will walk you from its leading minors to its signature, its principal axes, and what that shape means for the physics it encodes.

What is the actual difference between a bilinear form and a linear operator?

Both become matrices once a basis is fixed, so they are easy to confuse, but they are different objects. A linear operator maps a vector to a vector, $x\mapsto Ax$ — one input, one output, i.e. one upper and one lower index. A bilinear form maps two vectors to a scalar, $(x,y)\mapsto x^\mathsf{T}Ay$ — two inputs, no vector output, i.e. two lower indices. The consequence shows up under a change of basis: an operator's matrix transforms by similarity $P^{-1}AP$ , while a form's transforms by congruence $P^\mathsf{T}AP$ because both slots carry a factor of $P$ . They coincide only when $P$ is orthogonal — exactly the situation the principal-axis theorem exploits.

Why can I throw away the antisymmetric part of the matrix?

Because it contributes nothing to the quadratic form. Split $A=S+K$ with $S=\tfrac12(A+A^\mathsf{T})$ symmetric and $K=\tfrac12(A-A^\mathsf{T})$ antisymmetric. Now $x^\mathsf{T}Kx$ is a $1\times 1$ scalar, so it equals its own transpose: $x^\mathsf{T}Kx=(x^\mathsf{T}Kx)^\mathsf{T}=x^\mathsf{T}K^\mathsf{T}x=-x^\mathsf{T}Kx$ , using $K^\mathsf{T}=-K$ — a number equal to its own negative must be zero. So $Q(x)=x^\mathsf{T}Ax=x^\mathsf{T}Sx$ depends only on the symmetric part. This is why one always symmetrizes first: the antisymmetric part is invisible to $Q$ but would poison every eigenvalue and determinant test that assumes $A=A^\mathsf{T}$ . It is not useless in general — it describes an alternating form, the stuff of symplectic geometry — just not a quadratic form.

How do Lagrange reduction and the principal-axis method relate?

They are two ways to diagonalize the same form, answering different questions. Lagrange reduction (completing the square) uses whatever triangular, non-orthogonal substitution is convenient; it is fast, needs no eigenvalues, and delivers coefficients correct only in sign — you get the signature, nothing more. The principal-axis method uses the orthogonal matrix of eigenvectors; it is more work, but because the transformation is a rotation it preserves lengths and angles, so its diagonal entries are the genuine eigenvalues that set the true semi-axes. Rule of thumb: to classify a form — stable or unstable, ellipse or hyperbola — Lagrange is enough; for the actual geometry — principal axes, semi-axis lengths, normal-mode frequencies — you need the principal-axis route.

What does the signature physically mean?

The signature $(n_+,n_-,n_0)$ counts directions by the sign of the form along them. For a potential's Hessian, $n_+$ counts directions of upward curvature (restoring, stable) and $n_-$ downward (runaway, unstable): a positive-definite Hessian is a strict minimum, an indefinite one a saddle. For a metric, the signature separates Riemannian geometry (all-positive) from Lorentzian (one sign flipped, a light cone). For a covariance matrix, $n_0>0$ signals directions of zero variance — exact linear dependence in the data. In every case the signature is the coordinate-free summary: Sylvester's law guarantees no smooth invertible re-labeling of coordinates can change it.

Why is Cholesky preferred over eigenvalues to test definiteness numerically?

Speed and stability. Computing all eigenvalues of an $n\times n$ symmetric matrix is an iterative process costing on the order of $n^3$ operations with a large constant. A Cholesky factorization is a single direct pass, also $O(n^3)$ but roughly a factor of ten cheaper, and it either succeeds — proving positive-definiteness and handing you $L$ for free to solve linear systems — or it fails at the first nonpositive pivot, immediately certifying that the matrix is not positive-definite. That failure is itself the test. This is why optimization codes attempt a Cholesky of the Hessian at each iteration: success means "descend confidently, here is the Newton-step factor", failure means "this is a saddle, switch strategies". You get the verdict and the solver's workhorse in one operation.

Is the metric of spacetime really just a quadratic form?

Yes — the Minkowski metric is exactly a quadratic form, and a simple one. The invariant interval $Q=c^2t^2-x^2-y^2-z^2$ has matrix $\eta=\operatorname{diag}(1,-1,-1,-1)$ and signature $(1,3)$ (or $(3,1)$ in the opposite convention). Special relativity is, algebraically, the study of the linear maps that preserve this form — the Lorentz group, defined by $\Lambda^\mathsf{T}\eta\Lambda=\eta$ , is nothing but the congruence group of $\eta$ . Sylvester's law makes the theory consistent: no observer's coordinates can change the signature, so the split between "timelike" and "spacelike", and the light cone $Q=0$ between them, is absolute. In general relativity the metric becomes a field of quadratic forms, one per tangent space, and the demand that the signature stay $(1,3)$ everywhere is what makes the spacetime Lorentzian and causality well defined.