m=({{x, 1, 0, 0, 0, 1, 1, 0}, {1, x, 1, 0, 0, 0, 0, 0}, {0, 1, x, 1, 0, 0, 0, 0}, {0, 0, 1, x, 1, 0, 0, 1}, {0, 0, 0, 1, x, 1, 0, 0}, {1, 0, 0, 0, 1, x, 0, 0}, {1, 0, 0, 0, 0, 0, x+1, 0}, {0, 0, 0, 1, 0, 0, 0, x+1}})

Det[m]

Solve[-3-14 x-3 (x^2)+26 (x^3)+12 (x^4)-14 (x^5)-7 (x^6)+2 (x^7)+(x^8)==0, x]

n=({{0, 1, 2^.5, 0, 0, 0, 0, 0}, {1, 1, 0, 0, 0, 0, 0, 0}, {2^.5, 0, 1, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 1, 2^.5, 0, 0}, {0, 0, 0, 1, 1, 0, 0, 0}, {0, 0, 0, 2^.5, 0, -1, 0, 0}, {0, 0, 0, 0, 0, 0, -1, 0}, {0, 0, 0, 0, 0, 0, 0, 1}})

Eigenvalues[n]

Eigenvectors[n]