Use this Quadratic Form Calculator to enter values, adjust options, and review results in a compact responsive workspace.
Results are calculated automatically as you enter data.
If the matrix is not exactly symmetric, the calculator uses (A + A^T) / 2 because x^T A x only depends on the symmetric part.
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A quadratic form is a polynomial expression in which every term has total degree \(2\). In linear algebra, the most useful way to write it is as a matrix expression:
Here, \(\mathbf{x}\) is a vector of variables and \(A\) is the matrix that represents the form. For a two-variable form, \(\mathbf{x} = [x \; y]^T\); for a three-variable form, \(\mathbf{x} = [x \; y \; z]^T\).
Quadratic forms matter because they describe whether an expression is always positive, always negative, flat in some directions, or positive in some directions and negative in others. This is called the definiteness of the form. Definiteness is a central idea in linear algebra, multivariable calculus, optimization, statistics, physics, and engineering.
For example, the form \(x^2 + y^2\) is always positive except at the origin, so it is positive definite. The form \(x^2 - y^2\) can be positive or negative depending on the direction, so it is indefinite.
Definiteness turns a formula into geometric information. It tells you the basic shape of a quadratic expression:
Students often meet this idea when using the second derivative test in multivariable calculus. The Hessian matrix of a function creates a local quadratic approximation, and the definiteness of that quadratic form helps decide whether a critical point behaves like a minimum, maximum, saddle point, or a flatter case that needs more analysis.
Quadratic forms also appear in least-squares problems, covariance matrices, energy expressions, conic sections, ellipsoids, and stability analysis. In each case, the signs of the eigenvalues help explain the behavior of the expression.
A real quadratic form is easiest to study through a symmetric matrix. For two variables, a symmetric matrix has the form:
Then:
This is why a cross term is split between the two symmetric off-diagonal entries. If the expression contains \(4xy\), then the matching matrix entries are \(2\) and \(2\), not \(4\) and \(4\):
A non-symmetric matrix can represent the same quadratic form as its symmetric part:
So the sign behavior of a real quadratic form is controlled by a symmetric matrix.
The main reason eigenvalues are so useful is that every real symmetric matrix can be orthogonally diagonalized. In practical terms, this means the variables can be rotated or reflected into new coordinates where the cross terms disappear.
If \(A\) has eigenvalues \(\lambda_1, \lambda_2, \ldots, \lambda_n\), the quadratic form can be rewritten as:
The variables \(u_1, u_2, \ldots, u_n\) are coordinates measured along eigenvector directions. Since each squared term \(u_i^2\) is never negative, the sign of each coefficient \(\lambda_i\) determines the sign behavior of the whole form.
| Eigenvalue pattern | Definiteness classification | Meaning |
|---|---|---|
| All \(\lambda_i > 0\) | Positive definite | The form is positive for every nonzero vector. |
| All \(\lambda_i < 0\) | Negative definite | The form is negative for every nonzero vector. |
| All \(\lambda_i \ge 0\), with at least one zero | Positive semidefinite | The form is never negative, but may be zero away from the origin. |
| All \(\lambda_i \le 0\), with at least one zero | Negative semidefinite | The form is never positive, but may be zero away from the origin. |
| Both positive and negative eigenvalues | Indefinite | The form takes both positive and negative values. |
Principal minors give a determinant-based way to inspect a matrix. The leading principal minors are commonly written as:
For a real symmetric matrix, Sylvester's criterion says the matrix is positive definite exactly when all leading principal minors are positive:
Negative definiteness can be checked by applying the same positive-definite test to \(-A\). This produces alternating signs in the leading minors of \(A\).
Principal minors connect determinant information with definiteness. This calculator uses their exact signs for the authoritative classification, avoiding the scale-dependent zero thresholds that can affect numerical eigenvalues.
Consider:
The cross-term coefficient \(4\) is split across the symmetric off-diagonal entries:
For a \(2 \times 2\) symmetric matrix, the eigenvalues can be found from the trace and determinant. Here, the eigenvalues are:
Both values are positive, so the quadratic form is positive definite. Geometrically, it behaves like an upward-opening bowl after a rotation of coordinates.
Consider:
Its matrix is already diagonal:
The eigenvalues are \(1\) and \(-1\). Because one is positive and the other is negative, the form is indefinite.
You can see this directly from the expression. Along the \(x\)-axis, \(q(x,0)=x^2\), which is positive for \(x \ne 0\). Along the \(y\)-axis, \(q(0,y)=-y^2\), which is negative for \(y \ne 0\).
Consider:
Expanding gives:
The symmetric matrix is:
The eigenvalues are \(2\) and \(0\). The form is never negative, but it is zero whenever \(x=y\):
Because there is a zero eigenvalue and no negative eigenvalue, the form is positive semidefinite rather than positive definite.
The most important result is the classification. It tells you whether the quadratic form is positive definite, negative definite, positive semidefinite, negative semidefinite, or indefinite.
Exact principal-minor signs determine the reported classification. The displayed approximate eigenvalues explain the corresponding diagonalized shape: positive coefficients give upward directions, negative coefficients give downward directions, and mixed signs give both.
The diagonalized form shows the same idea in a coordinate system where the cross terms have been removed:
The leading minors and full set of principal minors determine the calculator's exact definiteness classification. Eigenvalues and the diagonalized form are displayed as numerical supporting information and include a residual check.
The symmetric matrix shown in the result is the matrix actually used to classify the form. If a non-symmetric matrix was entered, the displayed symmetric matrix may differ from the entries typed into the input grid.
The contour preview gives a visual sense of the shape. For a \(2 \times 2\) form, it represents contours of \(q(x,y)\). For a \(3 \times 3\) form, it represents only the \(z=0\) slice, so it should not be read as a full three-dimensional picture.
One common mistake is putting the full cross-term coefficient into each off-diagonal entry. In a symmetric \(2 \times 2\) matrix, the off-diagonal entry \(b\) creates a cross term \(2bxy\). Therefore, a term like \(6xy\) corresponds to off-diagonal entries \(3\) and \(3\).
Another mistake is assuming that a non-symmetric matrix is used exactly as typed. For a real quadratic form, only the symmetric part affects \(\mathbf{x}^T A \mathbf{x}\). The skew-symmetric part cancels out of the quadratic expression.
Near-zero eigenvalues also need care. A value displayed as \(0\) or very close to zero may reflect numerical tolerance rather than an exact symbolic result. This matters most when distinguishing definite forms from semidefinite forms.
Users also sometimes expect a quadratic-expression input to accept linear terms, constants, parentheses, functions, or symbolic parameters. A quadratic form contains only degree-\(2\) terms such as \(x^2\), \(xy\), \(xz\), and \(yz\).
For \(3 \times 3\) forms, the contour preview can be misunderstood. It shows a two-dimensional slice, not the complete three-variable geometry.
Use quadratic form definiteness when you need to understand the sign or shape of a second-degree expression.
Common uses include:
Classification is exact for supported decimal and scientific-notation inputs. The calculator displays exact rational symmetric-matrix entries and principal minors, then uses their signs for classification. It does not provide symbolic eigenvalue radicals; eigenvalues and the diagonalized form are binary64 approximations.
Decimal and scientific-notation inputs are converted to exact rational values for determinants and classification. Approximate eigenvalues are computed after scaling the matrix. If an eigenvalue is below the reliable numerical resolution, the interface says so while keeping the exact principal-minor classification authoritative.
Only \(2 \times 2\) and \(3 \times 3\) forms are supported. Variables beyond \(x\), \(y\), and \(z\) are not supported. Expression input must use supported quadratic terms only; constants, linear terms, non-quadratic powers, parentheses, functions, and symbolic parameters are outside the supported input format.
Very large magnitudes can be difficult for numerical calculations. Inputs with absolute value greater than \(10^{100}\) are rejected, and non-finite values are not allowed.
Exact matrix entries and principal minors remain exact. Approximate eigenvalues are rounded to readable significant digits, while very small or very large values appear in scientific notation.
The graph is a visual preview, not a proof. Some level curves may not appear within the fixed viewing window, and the \(3 \times 3\) preview only shows the \(z=0\) slice.
For exams, proofs, engineering work, scientific modeling, or other high-stakes uses, double-check important results and confirm whether numerical tolerance, rounding, or exact symbolic requirements affect the conclusion.
Positive definite means the quadratic form is positive for every nonzero input vector. In eigenvalue terms, a real symmetric matrix is positive definite when all of its eigenvalues are positive.
A positive definite form is strictly positive for every nonzero vector. A positive semidefinite form is never negative, but it can equal zero for some nonzero vector. A zero eigenvalue is the usual sign that the form may be semidefinite rather than definite.
In a symmetric matrix, the entries \(A_{12}\) and \(A_{21}\) both contribute to the \(xy\) term. If both entries equal \(b\), their combined contribution is \(2bxy\). That is why a cross term like \(8xy\) corresponds to off-diagonal entries \(4\) and \(4\).
A non-symmetric matrix can appear in \(\mathbf{x}^T A \mathbf{x}\), but only its symmetric part affects the result. The equivalent symmetric matrix is \(\frac{A+A^T}{2}\), so definiteness is normally discussed for symmetric matrices.
A real symmetric matrix can be orthogonally diagonalized, so eigenvalue signs explain the geometry of the sum of squared transformed variables. This calculator reports classification from exact rational principal-minor signs; its displayed eigenvalues are binary64 approximations that support that interpretation.
For two variables, the contour preview shows level curves of \(q(x,y)\). For three variables, it shows only the \(z=0\) slice. It is useful for intuition, but the classification comes from the matrix calculations.
No. A quadratic form uses only degree-\(2\) terms. Terms such as \(x\), \(3\), \(x^3\), \(\sin(x)\), or parenthesized symbolic expressions are not part of the supported quadratic-form input.
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