Matrikspotentialer

Matrikspotentialer is a term that appears in some Swedish mathematical and applied contexts to describe potential-energy–like scalar functions that are defined by or associated with a matrix. In this usage, a matrix potential assigns a single real value to a vector input in a way that generalizes the notion of a potential energy from scalar forms to matrix-based constructions. The concept is closely tied to quadratic forms and to the way a matrix can encode curvature and energy in multivariate settings.

A common formalization is the quadratic form VA(x) = x^T A x for a square matrix A and a vector x. If A is real and symmetric, VA(x) is real-valued for all x, and its gradient is ∇VA(x) = 2Ax. If A is not symmetric, one typically uses the symmetric part (A + A^T)/2 to obtain a real potential function. When A is positive definite, VA is strictly convex, and VA has a unique global minimum at x = 0. The Hessian of VA is A + A^T, which is positive definite in the symmetric positive definite case, reflecting the curvature of the potential landscape.

In applications, matrix potentials often model energy-like quantities. For example, in mechanical systems with generalized coordinates, choosing A as a stiffness or mass matrix leads to a quadratic energy function. In electrical network theory, potentials at nodes can be described through quadratic forms involving a Laplacian-like matrix, yielding an expression such as (1/2)x^T Lx for energy. Matrix potentials also appear in stability analysis via Lyapunov functions of the form V(x) = x^T P x with P positive definite.

See also potential theory, quadratic form, Lyapunov function, Laplacian, and energy function.