Given a monic cubic x^3+bx^2+cx+d=0, with real b,c,d, we can translate the origin to produce x^3+3px+2q=0, where if p=0, we have a trivial cube root solution. If p/=0, we can scale x to produce one of the following 3 exclusive cases, with the parameter a real: x^3-3x-2cos(3a)=0, [sqrt(discriminant)=sqrt(108)isin(3a)] /* 3 real roots */ x^3-3x-2cosh(3a)=0, or [sqrt(discriminant)=sqrt(108)sinh(3a)] x^3+3x-2sinh(3a)=0. [sqrt(discriminant)=sqrt(108)cosh(3a)] These equations factor as follows: x^3-3x-2cos(3a) = (x-2cos(a))(x-2cos(a+2pi/3))(x-2cos(a-2pi/3)) x^3-3x-2cosh(3a) = (x-2cosh(a))(x+cosh(a)+isqrt(3)sinh(a))(x+cosh(a)-isqrt(3)sinh(a)) x^3+3x-2sinh(3a) = (x-2sinh(a))(x+sinh(a)+isqrt(3)cosh(a))(x+sinh(a)-isqrt(3)cosh(a)) We can represent the case with 3 real roots as the characteristic polynomial of the following tridiagonal symmetric matrix: [-cos(3a) sin(3a) 0 ] [ sin(3a) cos(3a) sqrt(2)] [ 0 sqrt(2) 0 ] The eigenvectors corresponding to these real eigenvalues are the columns of this orthogonal matrix: [ sin(a) sin(a+2pi/3) sin(a-2pi/3)] [ cos(a) cos(a+2pi/3) cos(a-2pi/3)]*sqrt(2/3) [1/sqrt(2) 1/sqrt(2) 1/sqrt(2) ]