Bounds on the extremal eigenvalues of positive definite matrices.

Abstract

The minimum and maximum eigenvalues of a positive de nite matrix are crucial to determining the condition number of linear systems. These can be bounded below and above respectively using the Gershgorin circle theorem. Here we seek upper bounds for the minimum eigenvalue and lower bounds for the maximum eigenvalue. Intervals containing the extremal eigenvalues are obtained for the special case of Toeplitz matrices. The theory of quadratic forms is discussed in detail as it is fundamental in obtaining these bounds.