Numerical Linear Algebra with Applications

Chapter 13

Important Special Systems

Abstract

This chapter discusses two important types of matrices, tridiagonal, and positive definite. A tridiagonal matrix A can be factored into the product of a unit lower triangular matrix L and an upper triangular matrix U whose superdiagonal is equal to that of A. Solving a tridiagonal linear system with this algorithm is a slightly faster than using the Thomas algorithm. The chapter introduces the symmetric positive definite matrix and develops some of its properties. In particular, it shows that a matrix is positive definite if and only if its eigenvalues are positive. Sylvester’s criterion is stated but not proved. Two necessary criteria are developed that allow one to show a matrix is not positive ...