ABSTRACT

A system of linear equations can be expressed in the form of a matrix equation. This chapter presents some basic concepts and results from linear and matrix algebra, and from finite-dimensional vector spaces. It discusses the following determinants in detail: the Jacobian, the Hessian, and the bordered Hessians. The main application of the Hessian is found in large-scale optimization problems that use Newton-type methods. The Jacobian determinant is composed of all the first-order partial derivatives of a system of equations, arranged in an ordered sequence. For functions of several variables, the second-order conditions that are sufficient for a local maximum or minimum can be expressed in terms of the sequence of principal minors of the Hessian. The order of a bordered Hessian is determined by the order of the principal minor being bordered.