Mathematica Issue: solving matrix equation AX=\lambdaBX Symbolically

Question

I'm new to Mathematica, and I'm trying to solve a matrix equation in a form as

AX = \lambda BX

Here, Aand B are 4*4 matrices in the following, \lambda is a value, Xis the eigenvector- 4*1 matrix.

A = {{a1 + b1,  c,  d, f},
     {c,  a2 + b2 , f , e},
     {d , f , a3 + b1 , c},
     { f,  e , c,  a4 + b2}}

B = {{1,  0,  0 , 0},
     {0,  1 , 0 , 0},
     {0 , 0 , -1 , 0},
     {0,  0 , 0,  -1}}

I would like to solve this matrix equation and get the symbolical solution for \lambda using a1,a2,a3,a4,b1,b2,c,d,e,f, etc.

It would be much grateful if anyone can tell me.

Best regards,

mike

Answer 1

See Wolfram: Matrix Computations - specifically the section 'Generalized Eigenvalues'.

For n×n matrices A, B the generalized eigenvalues are the n roots of its characteristic polynomial, p(𝛇) = det(A - 𝛇 B). For each generalized eigenvalue, λ ∊ λ(A, B), the vectors, 𝛇, that satisfy

A χ = λ B χ

are described as generalized eigenvectors.

Example using symbolic values:

matA = {{a11, a12}, {a21, a22}};
matB = {{b11, b12}, {b21, b22}};

Eigenvalues[{matA, matB}]

{(1/(2 (-b12 b21+b11 b22)))(a22 b11-a21 b12-a12 b21+a11 b22-Sqrt[(-a22 b11+a21 b12+a12 b21-a11 b22)^2-4 (-a12 a21+a11 a22) (-b12 b21+b11 b22)]),(1/(2 (-b12 b21+b11 b22)))(a22 b11-a21 b12-a12 b21+a11 b22+Sqrt[(-a22 b11+a21 b12+a12 b21-a11 b22)^2-4 (-a12 a21+a11 a22) (-b12 b21+b11 b22)])}

Eigenvectors[{matA, matB}]

...