A quadratic form is a mathematical expression that contains only quadratic terms, meaning terms that are raised to the second power. It is typically written as a polynomial in several variables, with the highest degree term being quadratic.
Quadratic forms have many real-world applications, such as in physics to describe the energy of a particle in a potential field, in engineering to model the trajectory of a projectile, and in economics to analyze production functions.
The type of a quadratic form is determined by its discriminant, which is the expression inside the square root when the form is written in its standard form. If the discriminant is positive, the form is positive definite; if it is negative, the form is negative definite; and if it is zero, the form is indefinite.
Quadratic forms can be represented by symmetric matrices, also known as quadratic forms matrices. The coefficients of the quadratic form correspond to the entries in the matrix, and the variables in the form correspond to the rows and columns of the matrix. This relationship allows for easier computation and analysis of quadratic forms.
Quadratic forms are commonly used in optimization problems, such as finding the maximum or minimum value of a function. By converting the function into a quadratic form, techniques from linear algebra and calculus can be used to find the optimal solution.