1,1

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 (mathar(AT)strw.leidenuniv.nl), Mar 22 2006

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.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Index entries for sequences related to binary matrices

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

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}]

(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

Cf. A055548, A055549.

Sequence in context: A093990 A156448 A157752 this_sequence A113087 A099729 A123117

Adjacent sequences: A055544 A055545 A055546 this_sequence A055548 A055549 A055550

nonn,more

Eric Weisstein (eric(AT)weisstein.com)

Entry revised by N. J. A. Sloane (njas(AT)research.att.com), Jan 15, 2004

a(5) from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Mar 15 2006

a(6) from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Mar 22 2006

Statement (c) corrected. - Max Alekseyev (maxale(AT)gmail.com), Oct 18 2008