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A091472 Number of n X n matrices with entries {0,1} that are diagonalizable over the complex numbers. 2
2, 12, 320, 43892 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

A matrix M is diagonalizable over a field F if there is an invertible matrix S with entries from F such that S^(-1) M S is diagonal.

An n X n matrix M is diagonalizable if and only if it has n linearly independent eigenvectors.

REFERENCES

R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge, 1988, Section 1.3.

LINKS

Table of n, a(n) for n=1..4.

Eric Weisstein's World of Mathematics, Diagonalizable Matrix

Index entries for sequences related to binary matrices

EXAMPLE

a(2) = 12: all except 00/10, 01/00, 11/01, 10/11.

MATHEMATICA

Needs["Utilities`FilterOptions`"] Options[DiagonalizableQ]={ Field->Complexes, ZeroTest->(RootReduce[ # ]===0&) };

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

DiagonalizableQ[m_List?MatrixQ, opts___] := Module[ { field=Field/.{opts}/.Options[DiagonalizableQ], eigenopts=FilterOptions[Eigenvectors, opts] }, Switch[field, Complexes, ComplexDiagonalizableQ[m, eigenopts], Reals, RealDiagonalizableQ[m, eigenopts] ] ]

Table[Count[Matrices[n], _?DiagonalizableQ], {n, 4}]

(* Second program: *)

a[n_] := Module[{M, iter, cnt=0}, M = Table[a[i, j], {i, 1, n}, {j, 1, n}]; iter = Thread[{Flatten[M], 0, 1}]; Do[If[DiagonalizableMatrixQ[M], cnt++], Evaluate[Sequence @@ iter]]; cnt];

Do[Print[n, " ", a[n]], {n, 1, 4}] (* Jean-Fran├žois Alcover, Dec 09 2018 *)

CROSSREFS

Cf. A091470, A091471.

Sequence in context: A260321 A094047 A300045 * A156518 A012727 A296622

Adjacent sequences:  A091469 A091470 A091471 * A091473 A091474 A091475

KEYWORD

nonn,more

AUTHOR

Eric W. Weisstein, Jan 12 2004

STATUS

approved

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Last modified January 18 20:57 EST 2019. Contains 319282 sequences. (Running on oeis4.)