Matrix (mathematics) - Applications - Analysis and Geometry

Analysis and Geometry

The Hessian matrix of a differentiable function ƒ: Rn → R consists of the second derivatives of ƒ with respect to the several coordinate directions, i.e.

It encodes information about the local growth behaviour of the function: given a critical point x = (x₁, ..., x_n), i.e., a point where the first partial derivatives of ƒ vanish, the function has a local minimum if the Hessian matrix is positive definite. Quadratic programming can be used to find global minima or maxima of quadratic functions closely related to the ones attached to matrices (see above).

Another matrix frequently used in geometrical situations is the Jacobi matrix of a differentiable map f: Rn → Rm. If f₁, ..., f_m denote the components of f, then the Jacobi matrix is defined as

If n > m, and if the rank of the Jacobi matrix attains its maximal value m, f is locally invertible at that point, by the implicit function theorem.

Partial differential equations can be classified by considering the matrix of coefficients of the highest-order differential operators of the equation. For elliptic partial differential equations this matrix is positive definite, which has decisive influence on the set of possible solutions of the equation in question.

The finite element method is an important numerical method to solve partial differential equations, widely applied in simulating complex physical systems. It attempts to approximate the solution to some equation by piecewise linear functions, where the pieces are chosen with respect to a sufficiently fine grid, which in turn can be recast as a matrix equation.