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Quadratic Form

A scalar-valued expression built from a vector and a matrix, x'Ax, generalizing a sum of squares to allow cross-terms weighted by the matrix's off-diagonal entries -- the algebraic form underlying sums of squares in ANOVA, the multivariate normal density's exponent, and the Mahalanobis distance.

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In Plain English

A quadratic form is what you get when you sandwich a matrix between a vector and its own transpose, and the result is always a single number, not another vector or matrix. It's the natural generalization of a plain sum of squares to situations where the different components aren't independent of each other, which is exactly why the exponent of the multivariate normal density, and Mahalanobis distance, both use quadratic forms built from the inverse covariance matrix rather than a simple sum of squared deviations.

Definition

For an $n \times n$ matrix $A$ and a vector $x \in R^{n}$ , the quadratic form is the scalar $Q (x) = x^{⊤} Ax = \sum_{i = 1}^{n} \sum_{j = 1}^{n} a_{ij} x_{i} x_{j}$ . When $A = I$ , the quadratic form reduces to the ordinary sum of squares $\sum_{i} x_{i}^{2}$ ; more generally, the sign behavior of $Q (x)$ across all nonzero $x$ classifies $A$ as positive definite, positive semi-definite, negative definite, or indefinite. Quadratic forms are central to statistics well beyond this classification role, every ANOVA sum of squares is a quadratic form in the observed data, the exponent of the multivariate normal density is the quadratic form $(x - μ)^{⊤} Σ^{- 1} (x - μ)$ , and the Mahalanobis distance is the square root of that same expression.

Formula

Q (x) = x^{⊤} Ax

Notation

Q (x)

—the quadratic form, a scalar

A

—the (typically symmetric) matrix defining the form

Worked Example

For $A = (2113)$ and $x = (1, 2)$ , the quadratic form expands to $Q (x) = 2 (1)^{2} + 2 (1) (1) (2) + 3 (2)^{2} = 2 + 4 + 12 = 18$ -- note the off-diagonal entry (1) contributes a cross-term $2 \times 1 \times x_{1} x_{2}$ , doubled because it appears symmetrically in both $a_{12}$ and $a_{21}$ .

Properties

Every ANOVA sum of squares (treatment, error, total) can be written as a quadratic form in the observed data vector, $y^{⊤} Ay$ for an appropriate idempotent matrix $A$ -- this quadratic-form representation is what underlies Cochran's theorem, which establishes the independence and chi-squared distributions of the pieces of an ANOVA decomposition.
The exponent of the multivariate normal probability density is the quadratic form $(x - μ)^{⊤} Σ^{- 1} (x - μ)$ -- because $Σ^{- 1}$ is positive definite, this quadratic form is always non-negative, and its square root is exactly the Mahalanobis distance, a variance-adjusted measure of how far a point sits from the distribution's mean.
Only the symmetric part of a matrix contributes to its quadratic form -- for any matrix $A$ , $x^{⊤} Ax = x^{⊤} (\frac{A + A ^{⊤}}{2}) x$ , which is why quadratic forms in statistics are, by convention, almost always written with a symmetric matrix in the first place.

At a Glance

FormulaQ(x) = x'Ax, always a single scalar

Special case A = Ireduces to the ordinary sum of squares, sum of x_i^2

Sign classifies A aspositive/negative (semi-)definite, or indefinite

Appears inANOVA sums of squares, the multivariate normal exponent, Mahalanobis distance

Used in These Tests

Mahalanobis Distance Test

Last updated September 8, 2026← Back to the Dictionary