Clear[X]
A = {{0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 1/μ}, {0, 0, 0, 
          0, -1/μ, 0}, {0, 0, 0, 0, 0, 0}, {0, 0, -1/ϵ, 
          0, 0, 0}, {0, 1/ϵ, 0, 0, 0, 0}}
MatrixForm[A]
MatrixForm[Eigensystem[A]]
{EV1, EV2, EV3, EV4, EV5, 
          EV6} = {
{0, 0, 0, 1, 0, 
            0}, 
{1, 0, 0, 0, 0, 
            0}, 
{0, -(Sqrt[ϵ])[E1, E2, E3, B1, B2, B3], 0, 0, 0, (Sqrt[μ])[E1, E2, E3, B1, B2, B3]}, 
{0, 
            0, (Sqrt[ϵ])[E1, E2, E3, B1, B2, B3], 
            0, (Sqrt[μ])[E1, E2, E3, B1, B2, B3], 
            0}, 
{0, (Sqrt[ϵ])[E1, E2, E3, B1, B2, B3], 
            0, 0, 0, (Sqrt[μ])[E1, E2, E3, B1, B2, B3]}, 
{0, 
            0, -(Sqrt[ϵ])[E1, E2, E3, B1, B2, B3], 
            0, (Sqrt[μ])[E1, E2, E3, B1, B2, B3], 
            0}
};
EVMX = 
        Transpose[{EV1, EV2, EV3, EV4, EV5, 
            EV6}];
MatrixForm[EVMX]
Det[EVMX]

Dop[X_] := 
      Transpose[{(∂_E1) X, (∂_E2) X, (∂_E3) 
            X, (∂_B1) X, (∂_B2) X, (∂_B3) 
            X}]

LB[X_, Y_] := 
      Dop[X] . Y - Dop[Y] . X

LB12 = LB[EV2, EV1];
LB13 = LB[EV3, EV1];
LB14 = LB[EV4, EV1];
LB15 = LB[EV5, EV1];
LB16 = LB[EV6, EV1];

LB23 = LB[EV3, EV2];
LB24 = LB[EV4, EV2];
LB25 = LB[EV5, EV2];
LB26 = LB[EV6, EV2];

LB34 = LB[EV4, EV3];
LB35 = LB[EV5, EV3];
LB36 = LB[EV6, EV3];

LB45 = LB[EV5, EV4];
LB46 = LB[EV6, EV4];

LB56 = LB[EV6, EV5];

LBMX = {{0, LB12, LB13, LB14, LB15, LB16}, {-LB12, 0, LB23, LB24, 
            LB25, LB26}, {-LB13, -LB23, 0, LB34, LB35, 
            LB36}, {-LB14, -LB24, -LB34, 0, LB45, 
            LB46}, {-LB15, -LB25, -LB35, -LB45, 0, 
            LB56}, {-LB16, -LB26, -LB36, -LB46, -LB56, 
            0}};
MatrixForm[LBMX]

LBC12 = LinearSolve[EVMX, LB12];
LBC13 = LinearSolve[EVMX, LB13];
LBC14 = LinearSolve[EVMX, LB14];
LBC15 = LinearSolve[EVMX, LB15];
LBC16 = 
        LinearSolve[EVMX, 
          LB16];

LBC23 = LinearSolve[EVMX, LB23];
LBC24 = LinearSolve[EVMX, LB24];
LBC25 = LinearSolve[EVMX, LB25];
LBC26 = 
        LinearSolve[EVMX, 
          LB26];

LBC34 = LinearSolve[EVMX, LB34];
LBC35 = LinearSolve[EVMX, LB35];
LBC36 = 
        LinearSolve[EVMX, 
          LB36];

LBC45 = LinearSolve[EVMX, LB45];
LBC46 = 
        LinearSolve[EVMX, 
          LB46];

LBC56 = 
        LinearSolve[EVMX, 
          LB56];

LBCMX = {{0, LBC12, LBC13, LBC14, LBC15, LBC16}, {-LBC12, 0, 
            LBC23, LBC24, LBC25, LBC26}, {-LBC13, -LBC23, 0, LBC34, 
            LBC35, LBC36}, {-LBC14, -LBC24, -LBC34, 0, LBC45, 
            LBC46}, {-LBC15, -LBC25, -LBC35, -LBC45, 0, 
            LBC56}, {-LBC16, -LBC26, -LBC36, -LBC46, -LBC56, 0}};
MatrixForm[LBCMX]