Eigenvector are different for eig(A)=eig(A,I) {I is identity matrix} but eig(K/M,M/M) not equal to eig(K,M)

I am trying to fully understand the eigenvalue/eigenvector problem but I run into the following inconsistency with Matlab eig command

clear all
clc
m=1;
k=1;
M=[m 0;0 2*m];
K=[2*k -k;-k 3*k];
A=K/M
% Following 3 are identical as expected

[vect val]=eig(A)
[vect val]=eig(A,eye(2))
[vect,val]=eig(K/M,M/M)

% However, the below should be identical to case eig(K/M,M/M) but I get same eigenvalues
% but different eigenvectors
[vect,val]=eig(K,M)

Answer 1

[John D'Errico]

Ok, looking at what you did, lets see...

The generalized eigenvalue problem solves

A*X = lambda*B*X

where A and B are given matrices, X a vector, and lambda an eigenvalue.

If the matrix B has an inverse, then we can write this as:

(inv(B)*A)*X = lambda*X

Now, what does slash do? From, the help for slash:

/   Slash or right division.
    B/A is the matrix division of A into B, which is roughly the
    same as B*INV(A) , except it is computed in a different way.

So K/M is equivalent to K*inv(M).

Matrix multiplication is NOT commutative!!!!!!!!!!! Well, generally not. There are specific cases where you get lucky.

K*inv(M) is NOT the same as inv(M)*K.

So these compute the same sets of eigenvectors, subject to an arbitrary scaling of the vectors. The order of the eigenvalues was swapped too.

[vect,val]=eig(M\K,M\M)
vect =
                         1                         1
                      -0.5                         1
val =
                       2.5                         0
                         0                         1

[vect,val]=eig(K,M)
vect =
        -0.577350269189626        -0.816496580927726
        -0.577350269189626         0.408248290463863
val =
                         1                         0
                         0                       2.5