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
Eric Weisstein's World of Mathematics, Diagonalizable Matrix
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-François Alcover, Dec 09 2018 *)
PROG
(Sage)
import itertools
def a(n):
ans, W = 0, itertools.product([0, 1], repeat=n*n)
for w in W:
if Matrix(QQbar, n, n, w).is_diagonalizable(): ans += 1
return ans # Robin Visser, Sep 24 2023
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Eric W. Weisstein, Jan 12 2004
EXTENSIONS
a(5) from Robin Visser, Sep 24 2023
STATUS
approved