Quadratic Form Calculator
Analyze quadratic forms, determine definiteness, and find canonical form
About this calculator
A quadratic form calculator analyzes mathematical expressions of the form x^T A x, where A is a symmetric matrix and x is a variable vector. This powerful tool determines the definiteness properties (positive definite, negative definite, indefinite, or semidefinite) of quadratic forms and converts them to canonical form. It's essential for optimization problems, stability analysis in control theory, and understanding the geometric properties of quadratic surfaces in multivariable calculus and linear algebra applications.
How to use
Enter your symmetric matrix A or the coefficients of your quadratic form expression. The calculator will compute the eigenvalues to determine definiteness and perform diagonalization to find the canonical form. Review the results showing the definiteness classification, eigenvalues, and the transformed canonical representation of your quadratic form.
Frequently asked questions
What does positive definite mean in quadratic forms?
A quadratic form is positive definite when all eigenvalues are positive, meaning the form always produces positive values for non-zero input vectors.
How is canonical form different from standard form?
Canonical form eliminates cross-product terms through diagonalization, expressing the quadratic form as a sum of squares with eigenvalues as coefficients.
Can this calculator handle complex matrices?
The calculator works with real symmetric matrices, which are the standard form for real quadratic forms in most practical applications.