Quadratic Form Classifier

Math Linear Algebra • Applications and Advanced Linear Algebra (capstone)

Written by STEM Calculators Team

Classify the quadratic form \(q(x)=x^{\mathsf T}Ax\) using the eigenvalues (signature) of a symmetric matrix \(A\), and optionally check positive/negative definiteness using Sylvester’s criterion.

Matrix size

Method

Tolerance

Display decimals

Matrix \(A\)

Best results for symmetric \(A\). If not symmetric, the calculator will also show the symmetrized \(\tfrac12(A+A^{\mathsf T})\).

Inputs accept -3.5, 2e-4, fractions 7/3, and constants pi, e.

Ready

Results

Classification

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Signature \((n_+, n_-, n_0)\)

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Eigenvalues

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Trace / Determinant

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Symmetry check

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Diagonalization (symmetric case)

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Sylvester leading principal minors

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Graph

2×2 only. Plots a level set \(x^{\mathsf T}Ax = k\) (ellipse/hyperbola/lines) with axis tick values. Zoom with wheel/trackpad, drag to pan, double-click to reset. Play animates drawing the curve.

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Step-by-step

Enter \(A\) and click “Calculate”.

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Theory

Quadratic forms and definiteness

A quadratic form on \(\mathbb{R}^n\) is a function of the form \[ q(x)=x^{\mathsf T}Ax, \] where \(A\in\mathbb{R}^{n\times n}\). When \(A\) is symmetric, \(q(x)\) is purely “quadratic” (no ambiguity from cross terms), and it has clean geometric and optimization interpretations. In particular, the sign of \(q(x)\) for nonzero \(x\) determines the definiteness of the form:

Positive definite: \(q(x) > 0\) for all \(x\neq 0\).
Negative definite: \(q(x) < 0\) for all \(x\neq 0\).
Positive semidefinite: \(q(x) \ge 0\) for all \(x\) and \(q(x)=0\) for some nonzero \(x\).
Negative semidefinite: \(q(x) \le 0\) for all \(x\) and \(q(x)=0\) for some nonzero \(x\).
Indefinite: \(q(x)\) takes both positive and negative values.

Many problems in calculus and machine learning reduce to studying a quadratic form because the Hessian matrix \(H\) of a twice-differentiable function \(f\) appears in the local second-order approximation: \[ f(x+h)\approx f(x) + \nabla f(x)^{\mathsf T}h + \tfrac12\, h^{\mathsf T}Hh. \] If the Hessian is positive definite at a critical point, you have a strict local minimum; if negative definite, a strict local maximum; if indefinite, a saddle point.

Eigenvalues, signature, and diagonalization

For symmetric matrices, the spectral theorem states that \(A\) can be diagonalized by an orthonormal basis of eigenvectors: \[ A = P D P^{\mathsf T}, \] where \(P\) is orthogonal (its columns are eigenvectors) and \(D=\mathrm{diag}(\lambda_1,\dots,\lambda_n)\) contains real eigenvalues. Substituting into the quadratic form and using \(u=P^{\mathsf T}x\) gives \[ q(x) = x^{\mathsf T}PDP^{\mathsf T}x = u^{\mathsf T}Du = \sum_{i=1}^n \lambda_i u_i^2. \] Since each \(u_i^2\ge 0\), the sign behavior of \(q(x)\) is governed entirely by the signs of the eigenvalues.

The triple \((n_+,n_-,n_0)\)—the counts of positive, negative, and zero eigenvalues—is called the signature (or inertia). It determines the classification:

Positive definite iff \(n_+=n\).
Negative definite iff \(n_-=n\).
Positive semidefinite iff \(n_-=0\) and \(n_0>0\).
Negative semidefinite iff \(n_+=0\) and \(n_0>0\).
Indefinite iff \(n_+>0\) and \(n_->0\).

In 2D, the level sets \(x^{\mathsf T}Ax = k\) connect directly to conic sections. If \(A\) is positive definite, the set \(x^{\mathsf T}Ax=1\) is an ellipse (an “ellipsoid” in higher dimensions). If \(A\) is indefinite, \(x^{\mathsf T}Ax=1\) is a hyperbola. If \(A\) is semidefinite, the level set can degenerate into parallel lines or may be empty depending on the chosen \(k\).

Sylvester’s criterion (leading principal minors)

Another classic definiteness test is Sylvester’s criterion, which applies to symmetric matrices. Let \(\Delta_k\) be the determinant of the top-left \(k\times k\) block (the leading principal minor). Then:

\(A\) is positive definite iff \(\Delta_k>0\) for all \(k=1,\dots,n\).
\(A\) is negative definite iff \((-1)^k\Delta_k>0\) for all \(k\) (equivalently \(\Delta_1<0,\Delta_2>0,\Delta_3<0,\dots\)).

Sylvester’s criterion gives a fast PD/ND check without computing eigenvalues. However, semidefinite cases are more delicate: testing semidefiniteness generally involves checking all principal minors (not just leading ones) or using eigenvalues with a tolerance. For that reason, many numerical tools use eigenvalue-based classification as the primary method and treat Sylvester as a supplementary check.

Frequently Asked Questions

Why does the calculator prefer symmetric matrices?

For quadratic forms, only the symmetric part contributes: x^T A x = x^T ((A + A^T)/2) x. Symmetric matrices guarantee real eigenvalues and clean classification.

What is the signature (n+, n-, n0)?

It counts the number of positive, negative, and near-zero eigenvalues of A. The signature determines whether the form is definite, semidefinite, or indefinite.

What does Sylvester’s criterion test?

For symmetric matrices, it tests positive definiteness by requiring all leading principal minors Δk to be positive, and negative definiteness by alternating signs.

Why might a negative definite form plot x^T A x = -1 instead of 1?

If A is negative definite then x^T A x is always negative for x ≠ 0, so the level set x^T A x = 1 is empty. Using -1 yields an ellipse-like set.