eigen values of genralised symbolic matrix

Can we find out eigen values of genralised symbolic matrix like AX = LBX?, where A and B are matrix and L is eigen value

[Torsten]

Eigenvalues are zeros of the polynomial equation p(x)=det(A-x*B)=0. Thus if A has at most size (4x4), you can in principle symbolically determine the eigenvalues. But I wouldn't recommend this since the expressions for the polynomial and its associated roots will be nasty.

Answer 1

[Yashika]

Thanks, given code looks more promising will give it a shot, also i tried following method,

I am getting matrix of n*n in which each element is inner product between tw function... for simplicity perpose i have truncated it to 2*2 and 3*3. Then i converted symbolic matrices into normal matrix using C = double(A) and D = double(B) (if A is my matrix), and then i use eig(C,D). This worked.

Answer 2

[John D'Errico]

How large is your matrix? Note that computing the eigenvalues of an nxn matrix is mathematically equivalent to a problem of finding the roots of a nth degree polynomial. Even in the generalized eigenvalue problem, that issue still must apply.

But for a polynomial of degree greater than 4, that is provably impossible to do in general for a polynomial of degree >= 5. (Talk to Abel & Ruffini if you have complaints. They are dead of course, so the compaints desk is hard to get to, and they never answer the phone.)

A problem of course is that sym/eig does not handle the generalized eigenvalue problem. But you could still do this:

A = sym(randi(5,[3,3]))
A =
[ 3, 3, 2]
[ 3, 1, 1]
[ 4, 2, 3]
syms x
A(1) = x;
B = sym(magic(3));
eig(inv(B)*A)
ans =
 (53*x)/1080 + (((x/720 - ((53*x)/360 - 103/360)^3/27 + ((x/60 - 7/30)*((53*x)/360 - 103/360))/6 - 11/720)^2 - (((53*x)/360 - 103/360)^2/9 - x/180 + 7/90)^3)^(1/2) - x/720 + ((53*x)/360 - 103/360)^3/27 - ((x/60 - 7/30)*((53*x)/360 - 103/360))/6 + 11/720)^(1/3) + (((53*x)/360 - 103/360)^2/9 - x/180 + 7/90)/(((x/720 - ((53*x)/360 - 103/360)^3/27 + ((x/60 - 7/30)*((53*x)/360 - 103/360))/6 - 11/720)^2 - (((53*x)/360 - 103/360)^2/9 - x/180 + 7/90)^3)^(1/2) - x/720 + ((53*x)/360 - 103/360)^3/27 - ((x/60 - 7/30)*((53*x)/360 - 103/360))/6 + 11/720)^(1/3) - 103/1080
 (53*x)/1080 - (((x/720 - ((53*x)/360 - 103/360)^3/27 + ((x/60 - 7/30)*((53*x)/360 - 103/360))/6 - 11/720)^2 - (((53*x)/360 - 103/360)^2/9 - x/180 + 7/90)^3)^(1/2) - x/720 + ((53*x)/360 - 103/360)^3/27 - ((x/60 - 7/30)*((53*x)/360 - 103/360))/6 + 11/720)^(1/3)/2 - (3^(1/2)*((((x/720 - ((53*x)/360 - 103/360)^3/27 + ((x/60 - 7/30)*((53*x)/360 - 103/360))/6 - 11/720)^2 - (((53*x)/360 - 103/360)^2/9 - x/180 + 7/90)^3)^(1/2) - x/720 + ((53*x)/360 - 103/360)^3/27 - ((x/60 - 7/30)*((53*x)/360 - 103/360))/6 + 11/720)^(1/3) - (((53*x)/360 - 103/360)^2/9 - x/180 + 7/90)/(((x/720 - ((53*x)/360 - 103/360)^3/27 + ((x/60 - 7/30)*((53*x)/360 - 103/360))/6 - 11/720)^2 - (((53*x)/360 - 103/360)^2/9 - x/180 + 7/90)^3)^(1/2) - x/720 + ((53*x)/360 - 103/360)^3/27 - ((x/60 - 7/30)*((53*x)/360 - 103/360))/6 + 11/720)^(1/3))*1i)/2 - (((53*x)/360 - 103/360)^2/9 - x/180 + 7/90)/(2*(((x/720 - ((53*x)/360 - 103/360)^3/27 + ((x/60 - 7/30)*((53*x)/360 - 103/360))/6 - 11/720)^2 - (((53*x)/360 - 103/360)^2/9 - x/180 + 7/90)^3)^(1/2) - x/720 + ((53*x)/360 - 103/360)^3/27 - ((x/60 - 7/30)*((53*x)/360 - 103/360))/6 + 11/720)^(1/3)) - 103/1080
 (53*x)/1080 - (((x/720 - ((53*x)/360 - 103/360)^3/27 + ((x/60 - 7/30)*((53*x)/360 - 103/360))/6 - 11/720)^2 - (((53*x)/360 - 103/360)^2/9 - x/180 + 7/90)^3)^(1/2) - x/720 + ((53*x)/360 - 103/360)^3/27 - ((x/60 - 7/30)*((53*x)/360 - 103/360))/6 + 11/720)^(1/3)/2 + (3^(1/2)*((((x/720 - ((53*x)/360 - 103/360)^3/27 + ((x/60 - 7/30)*((53*x)/360 - 103/360))/6 - 11/720)^2 - (((53*x)/360 - 103/360)^2/9 - x/180 + 7/90)^3)^(1/2) - x/720 + ((53*x)/360 - 103/360)^3/27 - ((x/60 - 7/30)*((53*x)/360 - 103/360))/6 + 11/720)^(1/3) - (((53*x)/360 - 103/360)^2/9 - x/180 + 7/90)/(((x/720 - ((53*x)/360 - 103/360)^3/27 + ((x/60 - 7/30)*((53*x)/360 - 103/360))/6 - 11/720)^2 - (((53*x)/360 - 103/360)^2/9 - x/180 + 7/90)^3)^(1/2) - x/720 + ((53*x)/360 - 103/360)^3/27 - ((x/60 - 7/30)*((53*x)/360 - 103/360))/6 + 11/720)^(1/3))*1i)/2 - (((53*x)/360 - 103/360)^2/9 - x/180 + 7/90)/(2*(((x/720 - ((53*x)/360 - 103/360)^3/27 + ((x/60 - 7/30)*((53*x)/360 - 103/360))/6 - 11/720)^2 - (((53*x)/360 - 103/360)^2/9 - x/180 + 7/90)^3)^(1/2) - x/720 + ((53*x)/360 - 103/360)^3/27 - ((x/60 - 7/30)*((53*x)/360 - 103/360))/6 + 11/720)^(1/3)) - 103/1080