Eigen value and eigen vector of symbolic block matrix

clc
syms a51 a53 a54 a56 a62 a65 a67 a68 a71 a73 a74 a76 a81 a83 a84 a86 e
A= zeros(4);
B=eyes(4);
C=[a51 0 a53 a54; 0 a62 0 0;a71 0 a73 a74;a81 0 a83 a84];
F=[ 0 a56 0 0;a65 0 a67 a68;0 a76 0 0;0 a76 0 0];

G=[A;B;C;D]
[V,D]=eig(A)

I want to find all eigen values and eigen vectors of block matrix G. If I used code [V,D]=eig(A), its takes lot of times and not return any result.

if there any code to solve this type of matrix in less time. plz help

;

[Paul]

Code does not run.

clc
syms a51 a53 a54 a56 a62 a65 a67 a68 a71 a73 a74 a76 a81 a83 a84 a86 e
A= zeros(4);
B=eyes(4);   % should this be eye(4)?
C=[a51 0 a53 a54; 0 a62 0 0;a71 0 a73 a74;a81 0 a83 a84];
F=[ 0 a56 0 0;a65 0 a67 a68;0 a76 0 0;0 a76 0 0];
G=[A;B;C;D]  % D is not defined at this point?

Also, A is 4x4, B is 4x4, and C is 4x4. I don't see how G can be square.

Matrix A is all zeros, so eig() should (and does) work fine.

[ASHA RANI]

sorry, codes are as follows

clc
syms a51 a53 a54 a56 a62 a65 a67 a68 a71 a73 a74 a76 a81 a83 a84 a86 e
A= zeros(4);
B=eye(4);   % should this be eye(4)?
C=[a51 0 a53 a54; 0 a62 0 0;a71 0 a73 a74;a81 0 a83 a84];
D=[ 0 a56 0 0;a65 0 a67 a68;0 a76 0 0;0 a86 0 0];
G=[A;B;C;D]  % D is not defined at this point?

I tried eig(G),...but it contibuously processing. It takes long time and show no output. plz help

[Paul]

G is 16 x 4. Are you sure G shouldn't be defined as

G = [A B;C D]

Interestingly enough, for G as defined in the code above:

syms a51 a53 a54 a56 a62 a65 a67 a68 a71 a73 a74 a76 a81 a83 a84 a86 e
A= zeros(4);
B=eye(4);
C=[a51 0 a53 a54; 0 a62 0 0;a71 0 a73 a74;a81 0 a83 a84];
D=[ 0 a56 0 0;a65 0 a67 a68;0 a76 0 0;0 a86 0 0];
G=[A;B;C;D]

G = 

eig(G)

ans = 

I have no idea what this means nor why it doesn't return an error, as eig() does for a non-square numeric input.

[ASHA RANI]

G=[A B; C D]

G is a 8*8 matrix.

plz help

[Paul]

syms a51 a53 a54 a56 a62 a65 a67 a68 a71 a73 a74 a76 a81 a83 a84 a86 e
A= zeros(4);
B=eye(4);
C=[a51 0 a53 a54; 0 a62 0 0;a71 0 a73 a74;a81 0 a83 a84];
D=[ 0 a56 0 0;a65 0 a67 a68;0 a76 0 0;0 a86 0 0];
G=[A B;C D]

G = 

Some matrices with specific block structures have known eigenvalues. I don't know if this matrix does, but you might want to search around for "eigenvalues of block matrices" or something like that and see if any hits show up for a 2x2 block matrix with zeros in the upper left and identity in the upper right.

It can be shown, I think, that for this matrix the characteristic polynomial can be reduced to a fourth order equation with the eigenvalues being the positive and negative square roots of the roots of that equation. However, that fourth order polynomial does not have a closed form expression for the roots.

[ASHA RANI]

Yes, for this matrix the characteristic polynomial can be reduced to a fourth order equation with the eigenvalues being the positive and negative square roots of the roots of that equation.

But how this result help to find expresion for eigen value of G and also eigen vector?

plz help

[Sulaymon Eshkabilov]

A = zero(4);

And you are trying to compute:

eig(A) that is equivalent of: eig(zero(4))

Answer 1

[Sulaymon Eshkabilov]

THere are a few errs that need to be fixed before computing eigen values and vectors.

syms a51 a53 a54 a56 a62 a65 a67 a68 a71 a73 a74 a76 a81 a83 a84 a86 e
A= zeros(4);
B=eye(4);
C=[a51 0 a53 a54; 0 a62 0 0;a71 0 a73 a74;a81 0 a83 a84];
F=[ 0 a56 0 0;a65 0 a67 a68;0 a76 0 0;0 a76 0 0];

G=[A, B;C, F]
[V,D]=eig(G)
doc eig   % Get some good help from MATLAB documentation library.

[ASHA RANI]

I tried a lot from mathworks doc , but dont get any idea.