# Eigenvalues and Eigenvectors of Symbolic Matrix

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SHUBHAM PATEL on 3 Dec 2019
Answered: Stephan on 4 Dec 2019
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);

Stephan on 4 Dec 2019
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.