Quadratic Form Calculator
Evaluate quadratic forms and analyze positive/negative definiteness of matrices
About this calculator
Our Quadratic Form Calculator evaluates quadratic expressions of the form x^T A x, where A is a symmetric matrix and x is a vector. This powerful tool analyzes the definiteness properties of matrices, determining whether they are positive definite, negative definite, or indefinite. Essential for optimization problems, economics, physics, and advanced mathematics, it helps students and professionals understand critical points, stability analysis, and matrix properties in multivariable calculus and linear algebra applications.
How to use
Enter your symmetric matrix A by specifying its dimensions and element values. Input the vector x or leave it as variables for symbolic evaluation. Click calculate to get the quadratic form result and definiteness analysis. The calculator will show eigenvalues, principal minors, and classify the matrix as positive definite, negative definite, indefinite, or semidefinite.
Frequently asked questions
What is a quadratic form in mathematics?
A quadratic form is an expression x^T A x where A is a symmetric matrix and x is a vector, representing homogeneous polynomial equations of degree two.
How do I determine if a matrix is positive definite?
A matrix is positive definite if all eigenvalues are positive, all principal minors are positive, or the quadratic form is always positive for non-zero vectors.
What applications use quadratic form analysis?
Quadratic forms are used in optimization, machine learning, statistics for covariance matrices, physics for energy functions, and economics for utility theory and cost analysis.