Eigenvalues and Eigenvectors of Symbolic Matrix

I have a symbolic matrix of which I want to get Eigenvalues and Eigenvectors. I want Eigenvalues and Eigenvectors in symbolic form.

syms E t 

H = [E -t -t -t -t 0 0 0 0;-t E 0 0 0 -t -t 0 0;-t 0 E 0 0 0 0 -t -t;-t 0 0 E 0 -t 0 -t 0; -t 0 0 0 E 0 -t 0 -t; 0 -t 0 -t 0 E 0 0 0; 0 -t 0 0 -t 0 E 0 0; 0 0 -t -t 0 0 0 E 0; 0 0 -t 0 -t 0 0 0 E];
eig(H);

Answer 1

[Stephan]

syms E t 
H = [E -t -t -t -t 0 0 0 0;-t E 0 0 0 -t -t 0 0;-t 0 E 0 0 0 0 -t -t;...
    -t 0 0 E 0 -t 0 -t 0; -t 0 0 0 E 0 -t 0 -t; 0 -t 0 -t 0 E 0 0 0;...
    0 -t 0 0 -t 0 E 0 0; 0 0 -t -t 0 0 0 E 0; 0 0 -t 0 -t 0 0 0 E];
[V,D] = eig(H)

gives:

V =
 
[  0, -1, -1,                               0,                               0,                               0,                               0,                                 2,                                 2]
[ -1,  0,  0, (E + 2^(1/2)*t)/(2*t) - E/(2*t), (E + 2^(1/2)*t)/(2*t) - E/(2*t), (E - 2^(1/2)*t)/(2*t) - E/(2*t), (E - 2^(1/2)*t)/(2*t) - E/(2*t), E/(2*t) - (E - 2*2^(1/2)*t)/(2*t), E/(2*t) - (E + 2*2^(1/2)*t)/(2*t)]
[ -1,  0,  0, E/(2*t) - (E + 2^(1/2)*t)/(2*t), E/(2*t) - (E + 2^(1/2)*t)/(2*t), E/(2*t) - (E - 2^(1/2)*t)/(2*t), E/(2*t) - (E - 2^(1/2)*t)/(2*t), E/(2*t) - (E - 2*2^(1/2)*t)/(2*t), E/(2*t) - (E + 2*2^(1/2)*t)/(2*t)]
[  1,  0,  0, E/(2*t) - (E + 2^(1/2)*t)/(2*t), (E + 2^(1/2)*t)/(2*t) - E/(2*t), E/(2*t) - (E - 2^(1/2)*t)/(2*t), (E - 2^(1/2)*t)/(2*t) - E/(2*t), E/(2*t) - (E - 2*2^(1/2)*t)/(2*t), E/(2*t) - (E + 2*2^(1/2)*t)/(2*t)]
[  1,  0,  0, (E + 2^(1/2)*t)/(2*t) - E/(2*t), E/(2*t) - (E + 2^(1/2)*t)/(2*t), (E - 2^(1/2)*t)/(2*t) - E/(2*t), E/(2*t) - (E - 2^(1/2)*t)/(2*t), E/(2*t) - (E - 2*2^(1/2)*t)/(2*t), E/(2*t) - (E + 2*2^(1/2)*t)/(2*t)]
[  0,  0,  1,                               0,                              -1,                               0,                              -1,                                 1,                                 1]
[  0,  1,  0,                              -1,                               0,                              -1,                               0,                                 1,                                 1]
[  0,  1,  0,                               1,                               0,                               1,                               0,                                 1,                                 1]
[  0,  0,  1,                               0,                               1,                               0,                               1,                                 1,                                 1]
 
 
D =
 
[ E, 0, 0,             0,             0,             0,             0,               0,               0]
[ 0, E, 0,             0,             0,             0,             0,               0,               0]
[ 0, 0, E,             0,             0,             0,             0,               0,               0]
[ 0, 0, 0, E + 2^(1/2)*t,             0,             0,             0,               0,               0]
[ 0, 0, 0,             0, E + 2^(1/2)*t,             0,             0,               0,               0]
[ 0, 0, 0,             0,             0, E - 2^(1/2)*t,             0,               0,               0]
[ 0, 0, 0,             0,             0,             0, E - 2^(1/2)*t,               0,               0]
[ 0, 0, 0,             0,             0,             0,             0, E - 2*2^(1/2)*t,               0]
[ 0, 0, 0,             0,             0,             0,             0,               0, E + 2*2^(1/2)*t]

See documentation for eig also in its symbolic version.

[Tutku Ilgin Özcan]

When I try this, it does not satisfy H*V = V*D. Do you have any idea why it is so?

[Torsten]

syms E t 
H = [E -t -t -t -t 0 0 0 0;-t E 0 0 0 -t -t 0 0;-t 0 E 0 0 0 0 -t -t;...
    -t 0 0 E 0 -t 0 -t 0; -t 0 0 0 E 0 -t 0 -t; 0 -t 0 -t 0 E 0 0 0;...
    0 -t 0 0 -t 0 E 0 0; 0 0 -t -t 0 0 0 E 0; 0 0 -t 0 -t 0 0 0 E];
[V,D] = eig(H)

V = 

D = 

simplify(H*V-V*D)

ans =