Note on Hadamard’s determinant theorem
1947; American Mathematical Society; Volume: 53; Issue: 6 Linguagem: Inglês
10.1090/s0002-9904-1947-08853-4
ISSN1088-9485
Autores Tópico(s)Mathematics and Applications
ResumoIntroduction.We shall call a square matrix A of order n an Hadamard matrix or for brevity an iî-matrix, if each element of A has the value ±1 and if the determinant of A has the maximum possible value w n/2 .It is known that such a matrix A is an iï-matrix [l] 1 if, and only if, AA'~nE n where A f is the transpose of A and E n is the unit matrix of order n.It is also known that, if an iï-matrix of order n > 1 exists, n must have the value 2 or be divisible by 4. The existence of an iî-matrix of order n has been proved [2,3] only for the following values of n>\\ (a) w = 2, (b) w = £*+ls~0 mod 4, p a prime, (c) n -m(p h -\-l) where m^2 is the order of an ü-matrix and p is a prime, (d) n = q(q -l) where q is a product of factors of types (a) and (b), (e) n = 172 and for n a product of any number of factors of types (a), (b), (c), (d) and (e).In this note we shall show that an iï-matrix of order n also exists when (f) n -q(q+3) where q and g+4 are both products of factors of types (a) and (b), (g) n = nin2(p h +l)p h , where Wi>l and w 2 >l are orders of i7-matrices and p is an odd prime, and (h) n -nin2in(tn+3) where Wi>l and W2>1 are orders of jff-matrices and m and ra+4 are both of the form p h + l, p an odd prime.It is interesting to note the presence of the factors tii and w 2 in the types (g) and (h) and their absence in the types (d) and (f).Thus, if p is a prime and £*+ls=0 mod 4, an iJ-matrix of order p h (p h +l) exists but, if p h + l =2 mod 4, we can only be sure of the existence of an iJ-matrix of order nitt2p h (p h +l) where tii>l and ti2>l are orders of iï-matrices.This is analogous to the simpler result that, iî p h + 1^0 mod 4 an ü-matrix of order p h + l exists but, if p h +lz=2 mod 4, we can only be sure of the existence of an i?-matrix of order n(p h +l) where n > 1 is the order of an iï-matrix.We shall denote the direct product of two matrices A and B by A -B and the unit matrix of order n by E w .Theorems on the existence of iï-matrices.If a symmetric ü-matrix of order m>\ exists, there exists an iï-matrix H of order m with the form "C£
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