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A055547 Number of normal n X n matrices with entries {0,1}. 3
2, 8, 68, 1124, 36112, 2263268, 281249824 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

A complex matrix M is normal if M^H M = M M^H, where H is conjugate transpose.

Let M be an n X n complex matrix with eigenvalues l_1, ..., l_n. The following are equivalent:

(a) M is normal;

(b) There is a unitary matrix U such that U^H M U is diagonal;

(c) Sum_{i,j = 1..n} |M_{i,j}|^2 = |l_1|^2 + ... + |l_n|^2; and

(d) M has an orthonormal set of n eigenvectors.

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, Mar 22 2006

REFERENCES

G. H. Golub and C. F. van Loan, Matrix Computations, Johns Hopkins, 1989, p. 336.

R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge, 1988, Section 2.5.

W. H. Press et al., Numerical Recipes, Cambridge, 1986; Chapter 11.

LINKS

Table of n, a(n) for n=1..7.

R. J. Mathar, C program

Georg Muntingh, Sage code for recursively computing higher entries

Eric Weisstein's World of Mathematics, Normal Matrix.

Index entries for sequences related to binary matrices

FORMULA

a(n) >= 2^[n*(n+1)/2] = A006125(n+1) because all symmetric binary matrices (which have n*(n+1)/2 independent elements) are normal. - R. J. Mathar, Mar 22 2006

MATHEMATICA

Options[NormalMatrixQ]={ ZeroTest->(#===0&) };

Matrices[n_, l_List:{0, 1}] := Partition[ #, n]&/@Flatten[Outer[List, Sequence@@Table[l, {n^2}]], n^2-1]

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} ]

Table[Count[Matrices[n, {0, 1}], _?NormalMatrixQ], {n, 4}]

PROG

(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, Mar 15 2006

CROSSREFS

Cf. A006125, A055548, A055549.

Sequence in context: A262479 A192550 A157752 * A113087 A322495 A239843

Adjacent sequences:  A055544 A055545 A055546 * A055548 A055549 A055550

KEYWORD

nonn,more

AUTHOR

Eric W. Weisstein

EXTENSIONS

Entry revised by N. J. A. Sloane, Jan 15 2004

a(5) from R. J. Mathar, Mar 15 2006

a(6) from R. J. Mathar, Mar 22 2006

Statement (c) corrected. - Max Alekseyev, Oct 18 2008

a(7) from Georg Muntingh, Feb 03 2014

STATUS

approved

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Last modified January 17 23:15 EST 2019. Contains 319251 sequences. (Running on oeis4.)