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Quadratic Forms

Quadratic-form definiteness checker. Enter a symmetric matrix to test positive/negative/semidefinite via minors or eigenvalues.

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Q = xᵀ A x, all λᵢ>0 ⇒ positive definite
space-separated entries, semicolon-separated rows, e.g. 1 2;3 4

📖 Tutorial｜Quadratic Forms

1. Definition
A quadratic form Q = xᵀAx uses a symmetric A. Definiteness is judged by the signs of eigenvalues.
2. Symbols
SymbolMeaning
xᵀAxquadratic form
λᵢeigenvalues of A
positive definiteall λᵢ > 0
3. How this tool works
  • All eigenvalues > 0 ⇒ positive definite;
  • All eigenvalues < 0 ⇒ negative definite;
  • Mixed signs or zeros ⇒ indefinite.
4. Steps
  1. Enter a symmetric A;
  2. Click Calculate for eigenvalues and definiteness.
5. Example
Example: A=2 0;0 3. Eigenvalues 2,3 are both positive, so xᵀAx is positive definite.
6. Pitfalls
A must be symmetric;
Zero eigenvalues mean semi-definite or indefinite;
Real symmetric matrices have real eigenvalues.

❓ FAQ｜Quadratic Forms

What is a positive definite quadratic form?
xᵀAx > 0 for all nonzero x, equivalent to all eigenvalues > 0.
How to test negative definite?
All eigenvalues < 0.
What does indefinite mean?
Eigenvalues have mixed signs (or zeros), so the form can be positive or negative.
Why must A be symmetric?
The matrix of a quadratic form is taken as its symmetric part.
How to test positive semi-definite?
All eigenvalues ≥ 0 and at least one is 0.