How do I find the eigenvalues and vectors of an equation not of form (A*x = b*x)?

I'm trying to solve a vibrations problem in which my eigenvalue equation is K*x = b*M*x, where K and M are matrices and b is a scalar. I can do this by hand for low dimensional problems, but it gets to be way too much after more than 2-3 degrees of freedom are introduced.

Is there a built in command to do this? K and M are symbolic matrices.

[Jeffrey Clark]

@Edward Walker, as indicated in the tips documentation the symbolic eig function does not support solving the generalized eigenvalue problem (with two input arguments). To solve the generalized eigenvalue problem, use the MATLAB eig function instead by converting the input matrices to a MATLAB numeric type. As in this varient:

[V,D,W] = eig(A,B) also returns full matrix W whose columns are the corresponding left eigenvectors, so that W'*A = D*W'*B

Answer 1

[Torsten]

Multiply be inv(M) and use eig as

lambda = eig(inv(M)*K)

Answer 2

[John D'Errico]

This is a classic problem in eigenvalues, caled the generalized eigenvalue problem. That is, if you want to solve the eigenproblem

A*x = lambda*B*x

then eig solves it for you, directly.

help eig

 EIG    Eigenvalues and eigenvectors.
    E = EIG(A) produces a column vector E containing the eigenvalues of 
    a square matrix A.
 
    [V,D] = EIG(A) produces a diagonal matrix D of eigenvalues and 
    a full matrix V whose columns are the corresponding eigenvectors  
    so that A*V = V*D.
  
    [V,D,W] = EIG(A) also produces a full matrix W whose columns are the
    corresponding left eigenvectors so that W'*A = D*W'.
 
    [...] = EIG(A,'nobalance') performs the computation with balancing
    disabled, which sometimes gives more accurate results for certain
    problems with unusual scaling. If A is symmetric, EIG(A,'nobalance')
    is ignored since A is already balanced.
 
    [...] = EIG(A,'balance') is the same as EIG(A).
 
    E = EIG(A,B) produces a column vector E containing the generalized 
    eigenvalues of square matrices A and B.
 
    [V,D] = EIG(A,B) produces a diagonal matrix D of generalized
    eigenvalues and a full matrix V whose columns are the corresponding
    eigenvectors so that A*V = B*V*D.
 
    [V,D,W] = EIG(A,B) also produces a full matrix W whose columns are the
    corresponding left eigenvectors so that W'*A = D*W'*B.
 
    [...] = EIG(A,B,'chol') is the same as EIG(A,B) for symmetric A and
    symmetric positive definite B.  It computes the generalized eigenvalues
    of A and B using the Cholesky factorization of B.
 
    [...] = EIG(A,B,'qz') ignores the symmetry of A and B and uses the QZ
    algorithm. In general, the two algorithms return the same result,
    however using the QZ algorithm may be more stable for certain problems.
    The flag is ignored when A or B are not symmetric.
 
    [...] = EIG(...,'vector') returns eigenvalues in a column vector 
    instead of a diagonal matrix.
 
    [...] = EIG(...,'matrix') returns eigenvalues in a diagonal matrix
    instead of a column vector.
 
    See also CONDEIG, EIGS, ORDEIG.

    Documentation for eig
       doc eig

    Other uses of eig

       codistributed/eig    gpuArray/eig    sym/eig    symbolic/eig
       DynamicSystem/eig

Do you see that one of the options allows you to provide TWO matrices? All you need to do is:

[V,D] = eig(K,M);

Of course, if the matrix M is non-singular, then it is equivalent to writing the problem as

inv(M)*K*x = lambda*x

So then you could use eig simply as

[V,D] = eig(inv(M)*K);

In general, it is better to avoid the matrix inverse computation, so just use the generalized eigenvalue solver you already have in the form of eig(K,M).