The quadratic form appears frequently in many forms of signal processing applications. It comes up in least squares problems and is a critical component of the expression for the multivariate Gaussian probability distribution. Here we start off looking at several properties of the quadratic form and study the characteristics of the multidimensional surfaces and curves associated with this structure. This leads into a discussion on the multi–variate Gaussian probability density function (pdf). We then go on to investigate the relationship between positive definiteness and the quadratic form. We then discuss several methods for computing a single eigenpair associated with a matrix, one of which is based on the Rayleigh quotient, which is closely related to the quadratic form.