Category: Linear Algebra

Definite integrals have been introduced in Section 2.13 as a way to compute the area below the curve corresponding to the graph of a function of one variable. If we consider a function (x, y) of two variables, there is no reason why we should not consider its surface plot and the volume below the surface, corresponding to a region D on… Continue reading Integrals in multiple dimensions

In Section 2.7 we defined the derivative of a function of a single variable as the limit of an increment ratio: If we have a function of several variables, we may readily extend the concept above by considering a point  and perturbing one variable at a time. We obtain the concept of a partial derivative with respect to a single… Continue reading Partial derivatives: gradient and Hessian matrix

In this section we extend some concepts that we introduced in the previous concerning calculus for functions of one variable. What we really need for what follows is to get an intuitive idea of how some basic concepts are generalized when we consider a function of multiple variables, i.e., a function f(x1, x2, …, xn) = f(x) mapping a… Continue reading CALCULUS IN MULTIPLE DIMENSIONS

We explore the connections between linear algebra and calculus. This is necessary in order to generalize calculus concepts to functions of several variables; since any interesting management problem involves multiple dimensions, this is a worthy task. The simplest nonlinear function of multiple variables is arguably a quadratic form: Denoting the double sum as  is typically preferred to ,… Continue reading QUADRATIC FORMS

In Section 3.4.3 we observed that a square matrix  is a way to represent a linear mapping from the space of n-dimensional vectors to itself. Such a transformation, in general, entails both a rotation and a change of vector length. If the matrix is orthogonal, then the mapping is just a rotation. It may happen, for a specific vector v and… Continue reading EIGENVALUES AND EIGENVECTORS

From a formal perspective, we may use matrix inversion to solve a system of linear equations: From a practical viewpoint, this is hardly advisable, as Gaussian elimination entails much less work. To see why, observe that one can find each column  of the inverse matrix by solving the following system of linear equations: Here, vector ej is a… Continue reading Determinant and matrix inversion

The determinant of a square matrix is a function mapping square matrices into real numbers, and it is an important theoretical tool in linear algebra. Actually, it was investigated before the introduction of the matrix concept. In Section 3.2.3 we have seen that determinants can be used to solve systems of linear equations by Cramer’s rule. Another… Continue reading DETERMINANT

In this section we explore the link between a basis of a linear space and the possibility of finding a unique solution of a system of linear equations Ax = b, where , , and . Here, n is the number of variables and m is the number of equations; in most cases, we have m = n, but we may try to generalize a bit. Recall that… Continue reading Matrix rank

The possibility of expressing a vector as a linear combination of other vectors, or lack thereof, plays a role in many settings. In order to do so, we must ensure that the set of vectors that we want to use as a building blocks is “rich enough.” If we are given a set of vectors ,… Continue reading Linear independence, dimension, and basis of a linear space

Consider a stylized economy with three possible future states of the world, as illustrated in Fig. 3.9. Say that three securities are available and traded on financial markets, with the following state-contingent payoffs: These vectors indicate, e.g., that asset 1 has a payoff 1 if state 1 occurs, a payoff 2 if state 2 occurs, and… Continue reading Spanning sets and market completeness