Page 106 - Determinants and Their Applications in Mathematical Physics

4.6 Hessenbergians 91

If a ij = 0 when j − i> 1, the triangular array of zero elements appears
in the top right-hand corner. H n can be expressed neatly in column vector
notation.
Let

, (4.6.2)
T
C jr = a 1j a 2j a 3j ...a rj O n−r
n
where O i represents an unbroken sequence of i zero elements. Then

H n = C 12 C 23 C 34 ... C n−1,n C nn . (4.6.3)

n
Hessenbergians satisfy a simple recurrence relation.
Theorem 4.20.
n−1
H n =(−1) n−1 (−1) p r+1,n H r , H 0 =1,
r
r=0
where

a ij a j,j−1 a j−1,j−2 ··· a i+2,i+1 a i+1,i , j > i
p ij =
a ii , j = i.
Proof. Expanding H n by the two nonzero elements in the last row,
H n = a nn H n−1 − a n,n−1 K n−1 ,
where K n−1 is a determinant of order (n − 1) whose last row also contains
two nonzero elements. Expanding K n−1 in a similar manner,
K n−1 = a n−1,n H n−2 − a n−1,n−2 K n−2 ,
where K n−2 is a determinant of order (n − 2) whose last row also contains
two nonzero elements. The theorem appears after these expansions are
repeated a sufficient number of times.
Illustration.

H 5 = C 12 C 23 C 34 C 45 C 55 = a 55 H 4 − a 54 C 12 C 23 C 34 C 54 ,

C 12 C 23 C 34 C 54 = a 45 H 3 − a 43 C 12 C 23 C 53 ,

C 12 C 23 C 53 = a 35 H 2 − a 32 C 12 C 52 ,

C 12 C 52 = a 25 H 1 − a 21 a 15 H 0 .
Hence,
H 5 = a 55 H 4 − (a 45 a 54 )H 3 +(a 35 a 54 a 43 )H 2
−(a 25 a 54 a 43 a 32 )H 1 +(a 15 a 54 a 43 a 32 a 21 )H 0
= p 55 H 4 − p 45 H 3 + p 35 H 2 − p 25 H 1 + p 15 H 0 .
Muir and Metzler use the term recurrent without giving a definition of the
term. A recurrent is any determinant which satisfies a recurrence relation.