Page 241 - 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

b ik u k = H i ,
k=1
where

i − 1
b ik =
k − 1 H i−k
and hence express u k as a Hessenbergian whose elements are the H i .
2. Prove that

n n
iq
A A ir,sq
H(A ,A )= a .
is
rj
A pj A pr,sj
pq
p=1 q=1
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 of π to 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 order n whose elements are mono-
mials consists of the sum of n! terms each with n factors and that many