Principles of Mathematics in Operations Research

4 Eigen Values and Vectors

In this chapter, we will analyze determinant and its properties, definition of

eigen values and vectors, different ways how to diagonalize square matrices

and finally the complex case with Hermitian, unitary and normal matrices.

4.1 Determinants

4.1.1 Preliminaries

Proposition 4.1.1 det A ^ 0 => A is nonsingular.

Remark 4.1.2 Is A — XI (where X is the vector of eigen values) invertible?

det(A - XI) =? 0

where det(A — XI) is a polynomial of degree n in X, thus it has n roots.

Proposition 4.1.3 (Cramer's Rule) Ax = b where A is nonsingular. Then,

the solution for the jth unknown is

_ det(A(j <- b))

j~ det ,4 '

where A(j <— b) is the matrix obtained from A by interchanging column j with

the right hand side b.

Proposition 4.1.4 det A = ± [product of pivots].

Proposition 4.1.5 \detA\ = Vol(P), where P=conv{Y^"=1eiai, e^ is the jth
unit vector} is parallelepiped whose edges are from rows of A. See Figure 4-1-

Corollary 4.1.6 |detvl| = H?=1 \a,i\.

Definition 4.1.7 Let det A"^1 = g^.