Trace and Determinant of a Linear Operator

In this chapter, we study the trace and determinant of an operator. In the first section, we define the trace of a linear operator T on a finite-dimensional vector space V in terms of the characteristic polynomial, χT (x), of the operator. We also define the trace of a square matrix. We then relate these two concepts of trace by proving that if T is an operator on the finite-dimensional vector space V and B is any basis of V , then the trace of the operator T and the trace of the matrixMT (B,B) are the same. In the course of this, we establish many of the properties of the trace. In the second section, we introduce the determinant of a linear operator T on a finite-dimensional vector space V , again in terms of the characteristic polynomial, χT (x), of the operator. We also define a determinant of a square matrix. We then relate these two by proving that if T is an operator on the finite-dimensional vector space V and B is any basis of V , then the determinant of the operator T and the determinant of the matrixMT (B,B) are the same. In the concluding section, we show how the determinant can be used to define an alternating n-linear form on an n-dimensional vector space and prove that this form is unique.