Page 233 - Determinants and Their Applications in Mathematical Physics

218 5. Further Determinant Theory

Once again the symbol ←↔ denotes that the columns are arranged in
reverse order.

Illustrations

1
B 2 (x)= ,
1 x
 
1
B 3 (x)=  1 2x  ,
13x 4x 2
1
 
1
2  ,
 3x 
1 5x 9x
B 4 (x)= 
16x 19x 2 27x 3
1
 
1
 4x 
1 7x 16x  .
 2 
B 5 (x)= 
1 9x 37x 64x
 2 3 
110x 55x 2 175x 3 256x 4
Since b ij is a function of n, B n is not a submatrix of B n+1 . Finally, define
a block matrix N 2n of order 2n as follows:

N 2n = O S n (x) , (5.6.23)
B n (x) A n
where A n =[α m ] n , as before.

Illustrations

1
 
1
 ,
 x 
1 α 0 α 1
N 4 = 
1 xα 1 α 2
1
 
1
 3x 
1 x
 2 
 x 
1 
N 6 = 
 α 0 α 1 α 2 
1 2x α 1 α 2 α 3
 
13x 4x 2 α 2 α 3 α 4
N 2n is symmetric only when n =2.
A subset of N 4 is: