Knowra Quadratic form Quadratic form A quadratic form is a homogeneous degree-two polynomial in one or more variables. Over fields of characteristic not two, it can be represented by a symmetric matrix.
Symmetric matrix : A square matrix equal to its transpose. Its entries encode the coefficients of a quadratic form over fields of characteristic not two.
Vector space : A set equipped with vector addition and scalar multiplication satisfying the vector space axioms. Quadratic forms assign values to vectors in a vector space.
Conic section : A curve formed by intersecting a plane with a cone, including ellipses, parabolas, and hyperbolas. Equations of conics are built from quadratic terms in two variables.
Linear form : A linear function from a vector space to its scalar field. A linear form has degree one, while a quadratic form has degree two.
Joseph-Louis Lagrange : An eighteenth-century mathematician whose work shaped analysis, mechanics, and number theory. His four-square theorem established a landmark result about integers represented by quadratic forms.
Bilinear form : A function of two vector arguments that is linear in each argument separately. Polarization recovers a bilinear form from a quadratic form when the field's characteristic is not two.
Field (mathematics) : A set with addition, subtraction, multiplication, and division by nonzero elements. The underlying field affects diagonalization and whether a symmetric matrix represents the form.
Least squares : A method for estimating parameters by minimizing the sum of squared residuals. Its objective function is a quadratic form in the residual vector.
Indefinite quadratic form : A quadratic form that takes both positive and negative values on nonzero vectors. It contrasts with positive-definite forms, which stay positive away from zero.
Carl Friedrich Gauss : A German mathematician whose work transformed number theory, geometry, and mathematical physics. His theory of binary quadratic forms linked composition laws with arithmetic.
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