5 - CHARACTERISTIC EQUATION OF A TRANSFORMATION AND QUADRATIC FORMS

Abstract

This chapter discusses the characteristic equation of a transformation and quadratic forms. A quadratic form is a homogeneous polynomial of degree two in a number of variables. Quadratic forms occupy a central place in various branches of mathematics, including number theory, linear algebra, group theory, differential geometry, differential topology, and Lie theory. The theory of quadratic forms and methods used in their study depend in a large measure on the nature of the coefficients, which can be real or complex numbers, rational numbers, or integers. In linear algebra, analytic geometry, and in the majority of applications of quadratic forms, the coefficients are real or complex numbers. In the algebraic theory of quadratic forms, the coefficients are elements of a certain field. In the arithmetic theory of quadratic forms, the coefficients belong to a fixed commutative ring, frequently the integers Z or the p-adic integers Zp.