%I A055547
%S A055547 2,8,68,1124,36112,2263268
%N A055547 Number of normal n X n matrices with entries {0,1}.
%C A055547 A complex matrix M is normal if M^H M = M M^H, where H is conjugate transpose.
%C A055547 Let M be an n X n complex matrix with eigenvalues l_1, ..., l_n. The 
               following are equivalent:
%C A055547 (a) M is normal;
%C A055547 (b) There is a unitary matrix U such that U^H M U is diagonal;
%C A055547 (c) Sum_{i,j = 1..n} |M_{i,j}|^2 = |l_1|^2 + ... + |l_n|^2; and
%C A055547 (d) M has an orthonormal set of n eigenvectors.
%C A055547 If a normal matrix M is split into the symmetric and antisymmetric matrices 
               M=A+S with S=(M+M^H)/2 and A=(M-M^H)/2, M^H the transpose of M, A 
               must be a generalized Tournament matrix. (For Tournament matrices 
               each row and each column sums to zero.) The "generalization" is that 
               zeros (indicating a tie between the players) may occur outside the 
               main matrix diagonal. A is therefore a member of the set of the antisymmetric 
               ternary matrices (elements -1,0,+1) counted in A007081(n), because 
               there is a 1-to-1 mapping of the Tournament matrix onto the labeled 
               edge-oriented Eulerian graphs. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 
               Mar 22 2006
%D A055547 G. H. Golub and C. F. van Loan, Matrix Computations, Johns Hopkins, 1989, 
               p. 336.
%D A055547 R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge, 1988, Section 
               2.5.
%D A055547 W. H. Press et al., Numerical Recipes, Cambridge, 1986; Chapter 11.
%H A055547 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               NormalMatrix.html">Link to a section of The World of Mathematics.</
               a>
%H A055547 <a href="Sindx_Mat.html#binmat">Index entries for sequences related to 
               binary matrices</a>
%F A055547 a(n) >= 2^[n*(n+1)/2]=A00625(n+1) because all symmetric binary matrices 
               (which have n*(n+1)/2 independent elements) are normal. a(n) >= 2^n*a(n-1) 
               because symmetric attachment of any binary vector, of which there 
               are 2^n, to a normal matrix of dimension n-1 as a new last row and 
               last column produces a normal matrix with dimension n. - R. J. Mathar 
               (mathar(AT)strw.leidenuniv.nl), Mar 22 2006
%t A055547 Options[NormalMatrixQ]={ ZeroTest->(#===0&) };
%t A055547 Matrices[n_, l_List:{0, 1}] := Partition[ #, n]&/@Flatten[Outer[List, 
               Sequence@@Table[l, {n^2}]], n^2-1]
%t A055547 NormalMatrixQ[a_List?MatrixQ, opts___] := Module[ { b=Conjugate@Transpose@a, 
               zerotest=ZeroTest/.{opts}/.Options[NormalMatrixQ] }, (zerotest/@And@@Flatten[a.b-b.a])||Dimensions[a]=={1\
               , 1} ]
%t A055547 Table[Count[Matrices[n, {0, 1}], _?NormalMatrixQ], {n, 4}]
%o A055547 (PARI) NormaQ(a,n) = { local(aT) ; aT=mattranspose(a) ; if( a*aT == aT*a,
               1,0) ; } combMat(no,n) = { local(a,noshif) ; a = matrix(n,n) ; noshif=no 
               ; for(co=1,n, for(ro=1,n, if( (noshif %2)== 1,a[ro,co] = 1, a[ro,
               co] = 0) ; noshif = floor(noshif/2) ; ) ) ; return(a) ; } { for (n 
               = 1, 5, count = 0; a = matrix(n,n) ; for( no=0,2^(n^2)-1, a = combMat(no,
               n) ; count += NormaQ(a,n) ; ) ; print(count) ; ) } - R. J. Mathar 
               (mathar(AT)strw.leidenuniv.nl), Mar 15 2006
%Y A055547 Cf. A055548, A055549.
%Y A055547 Sequence in context: A093990 A156448 A157752 this_sequence A113087 A099729 
               A123117
%Y A055547 Adjacent sequences: A055544 A055545 A055546 this_sequence A055548 A055549 
               A055550
%K A055547 nonn,more
%O A055547 1,1
%A A055547 Eric Weisstein (eric(AT)weisstein.com)
%E A055547 Entry revised by N. J. A. Sloane (njas(AT)research.att.com), Jan 15, 
               2004
%E A055547 a(5) from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Mar 15 2006
%E A055547 a(6) from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Mar 22 2006
%E A055547 Statement (c) corrected. - Max Alekseyev (maxale(AT)gmail.com), Oct 18 
               2008