Determinants and Their Applications in Mathematical Physics

226 5. Further Determinant Theory

which, after transposition, is equivalent to the stated result.

Exercises

1.Prove that

i

∑

k=1

bikuk=Hi,

where

bik=

(

i− 1

k− 1

)

Hi−k

and hence expressukas a Hessenbergian whose elements are theHi.

2.Prove that

H(A

is

,A

rj

)=

n

∑

p=1

n

∑

q=1

a

′

pq

∣

∣

∣

∣

A

iq

A

ir,sq

A

pj

A

pr,sj

∣

∣

∣

∣

5.8 Some Applications of Algebraic Computing

5.8.1 Introduction

In the early days of electronic digital computing, it was possible to per-

form, in a reasonably short time, long and complicated calculations with

real numbers such as the evaluation ofπto 1000 decimal places or the

evaluation of a determinant of order 100 with real numerical elements, but

no system was able to operate with complex numbers or to solve even the

simplest of algebraic problems such as the factorization of a polynomial or

the evaluation of a determinant of low order with symbolic elements.

The first software systems designed to automate symbolic or algebraic

calculations began to appear in the 1950s, but for many years, the only

people who were able to profit from them were those who had easy access

to large, fast computers. The situation began to improve in the 1970s and

by the early 1990s, small, fast personal computers loaded with sophisticated

software systems had sprouted like mushrooms from thousands of desktops

and it became possible for most professional mathematicians, scientists, and

engineers to carry out algebraic calculations which were hitherto regarded

as too complicated even to attempt.

One of the branches of mathematics which can profit from the use of

computers is the investigation into the algebraic and differential properties

of determinants, for the work involved in manipulating determinants of or-

ders greater than 5 is usually too complicated to tackle unaided. Remember

that the expansion of a determinant of ordernwhose elements are mono-

mials consists of the sum ofn! terms each withnfactors and that many