Common Errors With Quadratic Forms

Why Small Notational Errors Change the Answer

A quadratic form can be written as a polynomial or as a matrix expression, and moving between those representations requires care. A missing factor of two can change a positive definite form into an indefinite one. A nonsymmetric matrix can represent a quadratic form, but its eigenvalues do not in general classify that form. And a test using leading principal minors answers a specific question only when its hypotheses are met.

Throughout, the scalars are real. A quadratic form on \(\mathbb{R}^n\) is a function \(q:\mathbb{R}^n\to\mathbb{R}\) expressible as \(q(\mathbf{x})=\mathbf{x}^{\top}\mathbf{A}\mathbf{x}\) for some real \(n\times n\) matrix \(\mathbf{A}\). A form is positive definite if \(q(\mathbf{x})>0\) for every \(\mathbf{x}\ne\mathbf{0}\); it is positive semidefinite if \(q(\mathbf{x})\ge0\) for every \(\mathbf{x}\). Negative definite and negative semidefinite are defined by reversing the inequalities. A form is indefinite if it takes both positive and negative values.

When the matrix is symmetric, its entries have a direct polynomial interpretation: diagonal entries give square-term coefficients, and each off-diagonal entry contributes twice to the corresponding mixed term. That factor of two is the source of the first common error.

Mixed Terms and the Correct Matrix

The check is not merely cosmetic: using \(2\) rather than \(1\) in both off-diagonal entries would produce a coefficient of \(4\) on \(xy\), not \(2\). The resulting form would be different and could have a different definiteness classification.

What a Nonsymmetric Matrix Represents

A quadratic form may be presented using a nonsymmetric matrix, but its skew-symmetric part contributes nothing. The relevant matrix for classifying the form is the symmetric part \(\tfrac12(\mathbf{A}+\mathbf{A}^{\top})\).

Proof. The scalar \(\mathbf{x}^{\top}\mathbf{A}\mathbf{x}\) equals its transpose. Transposing this \(1\times1\) matrix gives \((\mathbf{x}^{\top}\mathbf{A}\mathbf{x})^{\top}=\mathbf{x}^{\top}\mathbf{A}^{\top}\mathbf{x}\). Therefore \(\mathbf{x}^{\top}\mathbf{A}\mathbf{x}=\mathbf{x}^{\top}\mathbf{A}^{\top}\mathbf{x}\). Adding these equal expressions and dividing by \(2\) proves the identity. \(\square\)

A common mistake is to classify a quadratic form by looking at the eigenvalues of the nonsymmetric matrix used to write it. The theorem shows why that is unsafe: the form is determined by the symmetric part, not by the full nonsymmetric matrix. The next example makes the distinction concrete.

Congruence Is Not Similarity

Changing variables in a quadratic form has a specific matrix rule. If \(\mathbf{x}=\mathbf{P}\mathbf{y}\), then

The matrix \(\mathbf{P}^{\top}\mathbf{A}\mathbf{P}\) is obtained by congruence. In contrast, a similarity transformation has the form \(\mathbf{P}^{-1}\mathbf{A}\mathbf{P}\) and describes a change of basis for a linear map. These formulas are generally different. They coincide in the orthogonal change of variables used in orthogonal diagonalization, because an orthogonal matrix satisfies \(\mathbf{P}^{-1}=\mathbf{P}^{\top}\); for a general invertible \(\mathbf{P}\), that equality does not hold.

Proof. For every \(\mathbf{y}\), the transformed form has value \(\mathbf{y}^{\top}\mathbf{P}^{\top}\mathbf{A}\mathbf{P}\mathbf{y}=(\mathbf{P}\mathbf{y})^{\top}\mathbf{A}(\mathbf{P}\mathbf{y})\). Because \(\mathbf{P}\) is invertible, \(\mathbf{P}\mathbf{y}\ne\mathbf{0}\) whenever \(\mathbf{y}\ne\mathbf{0}\), and every nonzero \(\mathbf{x}\) equals \(\mathbf{P}\mathbf{y}\) for the nonzero vector \(\mathbf{y}=\mathbf{P}^{-1}\mathbf{x}\). Thus the two forms have the same sign on corresponding nonzero vectors, in both directions. Replacing “positive” by “nonnegative,” “negative,” or “nonpositive” gives the other stated conclusions. \(\square\)

Leading Principal Minors: Check the Hypotheses

For an \(n\times n\) matrix \(\mathbf{A}\), its \(k\)th leading principal minor is \(\Delta_k\), the determinant of the submatrix in the first \(k\) rows and first \(k\) columns. For a real symmetric matrix, Sylvester’s criterion says that the form is positive definite if and only if \(\Delta_k>0\) for every \(k=1,\dots,n\). The symmetry hypothesis and the strict inequalities both matter. The criterion is not a test for positive semidefiniteness, and merely checking one leading minor cannot establish positive definiteness.

The opposite error is to apply the positive-definite test to a matrix that is not symmetric, even when the question concerns the quadratic form it represents. First replace that matrix by its symmetric part; then use a definiteness criterion with the correct hypotheses. For positive definiteness, the leading principal minors must all be strictly positive. For positive semidefiniteness, nonnegative leading principal minors alone are not enough, as the example shows.

A Reliable Diagnostic Routine

When a quadratic-form calculation seems inconsistent, separate the representation from the classification. Use this sequence of checks:

The safest final check is to evaluate the polynomial and the matrix expression on the same vector, or to substitute the proposed change of variables directly. A single carefully chosen vector can expose a false definiteness claim: if a purportedly positive semidefinite form takes a negative value, the claim is disproved. Conversely, one positive value does not prove positive definiteness; that requires a valid argument covering every nonzero vector.