Eigenvectors@Transpose[A]

SetAttributes[makeVecs, HoldAll];
makeVecs[vals_, vecs_, s_, rQ_] := With[{n = Length[vals]},
    If[TrueQ[rQ],
       Do[Switch[Sign[Im[vals[[k]]]], 0, Null,
                 1, vecs[[k]] += s I vecs[[k + 1]],
                 -1, vecs[[k]] = Conjugate[vecs[[k - 1]]]], {k, n}]]]

Options[getEigensystem] = {Mode -> Automatic};

getEigensystem[mat_?SquareMatrixQ, opts : OptionsPattern[]] :=
   Module[{m = mat, chk, ei, ev, lm, lv, n, rm, rQ, rv}, n = Length[mat];
          Switch[OptionValue[Mode],
                 Right | Automatic, {lm, rm} = {"N", "V"},
                 Left, {lm, rm} = {"V", "N"},
                 All, {lm, rm} = {"V", "V"},
                 _, {lm, rm} = {"N", "V"}];
          LinearAlgebra`LAPACK`GEEV[lm, rm, m, ev, ei, lv, rv, chk, rQ];
          If[! TrueQ[chk], Message[getEigensystem::eivec0]; Return[$Failed, Module]];
          If[rQ, ev += I ei];
          Switch[OptionValue[Mode],
                 Right | Automatic, rv = ConjugateTranspose[rv];
                                    makeVecs[ev, rv, 1, rQ]; {ev, rv},
                 Left, lv = ConjugateTranspose[lv];
                       makeVecs[ev, lv, -1, rQ]; {ev, lv},
                 All, {lv, rv} = ConjugateTranspose /@ {lv, rv};
                      makeVecs[ev, rv, 1, rQ]; makeVecs[ev, lv, -1, rQ];
                      {ev, rv, lv},
                 _, rv = ConjugateTranspose[rv];
                    makeVecs[ev, rv, 1, rQ]; {ev, rv}]]

m = {{1., -2., 1., 9., 6.},
     {8., 5., 3., 7., 7.},
     {-9., -3., 9., 5., -2.},
     {-5., -4., 6., 8., -6.},
     {-3., 9., -2., 6., -3.}};

{vals, rvecs, lvecs} = getEigensystem[m, Mode -> All];

Norm[m.Transpose[rvecs] - Transpose[rvecs].DiagonalMatrix[vals]]
   4.16257*10^-14

Norm[lvecs.m - DiagonalMatrix[vals].lvecs]
   2.73754*10^-14