Quadratic Form Calculator
This quadratic form calculator evaluates a two-variable quadratic form, a x squared plus b xy plus c y squared, at a point you choose, and classifies the form\u2019s definiteness. Enter the three coefficients and the point, and the calculator returns the value and says whether the form is positive definite, negative definite, indefinite or degenerate. Quadratic forms are central to optimisation, statistics and geometry, where positive definiteness signals a genuine minimum and indefiniteness a saddle. The classification uses the sign of a and the determinant of the symmetric matrix that represents the form. The method, a worked example and the assumptions are below.
The formula
The form is Q(x, y) = a x^2 + b xy + c y^2, represented by the symmetric matrix with rows (a, b/2) and (b/2, c), whose determinant is ac minus b^2/4. It is positive definite if a is positive and that determinant is positive, negative definite if a is negative and the determinant positive, and indefinite if the determinant is negative.
Worked example
For Q = x^2 + y^2 (a = 1, b = 0, c = 1) at (3, 4), the value is 9 + 16 = 25, the determinant is 1, and a is positive, so the form is positive definite. Enter 1, 0, 1, 3, 4 to confirm.
Frequently asked questions
What is a quadratic form?
A homogeneous degree two expression in the variables, here a x squared plus b xy plus c y squared.
What does positive definite mean?
The form is positive for every non-zero input, which corresponds to a genuine minimum in optimisation.
How is definiteness decided?
From the sign of a and the determinant of the symmetric matrix: both conditions together give definiteness; a negative determinant gives an indefinite (saddle) form.
Who this calculator is for
This calculator is for algebra students and anyone needing a quick, reliable result.
What this calculator assumes
- You enter valid numbers.
- The standard method is applied.
- Results are rounded for display.
Formula and sources
Related calculators
- Quadratic Formula Calculator: solve a quadratic.
- Scientific Calculator: general maths.
- Matrix Calculator: matrix operations.
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