1,1

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.

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

Eric Weisstein's World of Mathematics, Diagonalizable Matrix

Index entries for sequences related to binary matrices

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

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}]

Cf. A091470, A091471.

Sequence in context: A012422 A122767 A094047 this_sequence A156518 A012727 A088229

Adjacent sequences: A091469 A091470 A091471 this_sequence A091473 A091474 A091475

nonn,more

Eric Weisstein (eric(AT)weisstein.com), Jan 12, 2004