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I have a matrix with 16 elements:

matrix = {{f1, -f2, -f2, f3 }, {f4, -f1, -f1, f2}, {f4, -f1, -f1, f2}, {f5, -f4, -f4, f1}}

I would like to calculate eigenvectors and eigenstates of the matrix, with additional assumption, that is elements f1, f2, f3, f4 and f5 are orthogonal. I've tried to call $Assumptions = f1*f2=f2*3 ... =0, but without luck. Any suggestions appreciated!

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    $\begingroup$ What do you mean by "elements are orthogonal"? The $Assumptions line contains syntax errors so we can't use it to infer unambiguously what you actually want or tried. $\endgroup$ – Jens May 15 '16 at 15:52
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Guessing OP meant f1*f2 == f1*f3 == f1* f4 == f1*f5 == f2*f3 == f2*f4 == f2*f5 == f3*f4 == f3*f5== f4*f5 ==0,

matrix = {{f1, -f2, -f2, f3}, {f4, -f1, -f1, f2}, {f4, -f1, -f1, f2}, {f5, -f4, -f4, f1}};
assumptions = Thread[(Subsets[Times[f1, f2, f3, f4, f5], {2}]) == 0];
ToRadicals @ FullSimplify[Eigensystem[matrix], Assumptions -> assumptions]

Mathematica graphics

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