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5.6 Distinct Matrices with Nondistinct Determinants 213
Define a Turanian T nr (Section 4.9.2) as follows:
φ r−2n+2 ... φ r−n+2
. .
. . , r ≥ 2n − 2, (5.6.7)
T nr = . .
φ r−n+1 φ r
...
n
which is a subdeterminant of M.
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
... 1 ...
... 1 4x ...
... 1 3x 6x 2 ...
... 1 2x 3x 2 4x 3 ...
... 1 x x 2 x 3 x 4 ...
... 1 φ 0 φ 1 φ 2 φ 3 φ 4 ...
... 1 x φ 1 φ 2 φ 3 φ 4 φ 5 ...
... 1 2x x 2 φ 2 φ 3 φ 4 φ 5 φ 6 ...
... 1 3x 3x 2 x 3 φ 3 φ 4 φ 5 φ 6 φ 7 ...
... 14x 6x 2 4x 3 x 4 φ 4 φ 5 φ 6 φ 7 φ 8 ...
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
The infinite matrix M
. . . . . . . . . . . . . . . . . . . .
. . . . . . . . . .
... 1 ...
... 1 −4x ...
... 1 −3x 6x 2 ...
... 1 −2x 3x 2 −4x 3 ...
... 1 −x x 2 −x 3 x 4 ...
... 1 α 0 α 1 α 2 α 3 α 4 ...
... 1 −x α 1 α 2 α 3 α 4 α 5 ...
... 1 −2x x 2 α 2 α 3 α 4 α 5 α 6 ...
... 1 −3x 3x 2 −x 3 α 3 α 4 α 5 α 6 α 7 ...
... 1 −4x 6x 2 −4x 3 x 4 α 4 α 5 α 6 α 7 α 8 ...
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
The infinite matrix M ∗
∗
The element α r occurs (r + 1) times in M . Consider all the subdeter-
minants of M which contain the element α r in the bottom right-hand
∗
corner and whose order n is sufficiently large for them to contain the el-
ement α 0 but sufficiently small for them not to have either unit or zero
elements along their secondary diagonals. Denote these determinants by
B , s =1, 2, 3,.... Some of them are symmetric and unique whereas oth-
nr
s
ers occur in pairs, one of which is the transpose of the other. They are
