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 Dec 4 comment Do complex numbers really exist? @stevenvh: Exactly the same representation works, it's just harder to visualize a 4D space! Dec 4 comment Transcendental Equations, Matrix (Eigenvalue problem?) (Mathematica) In fact, the determinant is the sum of terms like $-k_1 \cosh(k_2 \alpha_r)\cosh(k_3 \alpha_r)\sinh(k_4 \alpha_r)\sinh(k_5 \alpha_r)$ (where each $k_i>0$ is different), so it is clearly negative for all real $\alpha_r \neq 0$. And is only zero for $\alpha_r = 0$. Dec 4 revised Transcendental Equations, Matrix (Eigenvalue problem?) (Mathematica) Cleaned up a bit, and used image for matrix, because it scales better. Dec 4 comment Transcendental Equations, Matrix (Eigenvalue problem?) (Mathematica) dunks: If you plot the determinant, you see that 0 is clearly the only solution, which agrees with the result of FindRoot. Maybe you have an error in your matrix? (Or maybe I had a copying error - I added the Mma code for the matrix to your post, can you check it?) Dec 4 suggested approved edit on Transcendental Equations, Matrix (Eigenvalue problem?) (Mathematica) Dec 3 comment How to obtain Eigenvalues with maximum precision in Mathematica @George: Yeah, I don't think that in this case the error is anything to worry about. You can compare Eigenvalues[N[{A, B}, 50]] with N[Eigenvalues[Inverse[B].A], 50] to check. Apart from possible differences in ordering, they should be the same up to a small numerical error. Dec 3 comment How to obtain Eigenvalues with maximum precision in Mathematica @J.M.: He hasn't got a good record on asking clear questions. (I probably should stop bothering trying to answer them...) Dec 3 comment How to obtain Eigenvalues with maximum precision in Mathematica Or use Eigenvalues[N[{M, A}, prec]] from the very beginning - and forget about exact expressions. Dec 3 comment How to obtain Eigenvalues with maximum precision in Mathematica @George: The Solve[CharacteristicPolynomial[...]] is essentially what Eigenvalues does when working with exact expressions. In general there will be no "nice" way of writing the roots of a 31st degree polynomial, so just use N[expr, prec] to get the numerical values to what ever precision you needed. Dec 3 comment How to obtain Eigenvalues with maximum precision in Mathematica @George: Just use N[Root[...], prec] to get the precision (prec) that you need. In general there will be no "nice" way of writing the roots of a 31st degree polynomial. Dec 3 comment How to obtain Eigenvalues with maximum precision in Mathematica I just looked at your matrices. The exact Eigenvalues[Inverse[A].M] works (took a couple of minutes on my machine) and yields Root objects of degree 31 with massive integer coefficients (~1000 digits). RootApproximant won't find these unless you use insane precision. A Root expression represents the abstract root of a polynomial. The order that they are defined in is defined in the documentation. They can be evaluated to arbitrary (within reason) precision using N[]. Dec 3 comment How to obtain Eigenvalues with maximum precision in Mathematica @J.M.: {Dimensions@#, MatrixRank@#} & /@ {A, B} returns {{{33, 33}, 31}, {{33, 33}, 33}}, so B is invertible. Dec 3 answered How to obtain Eigenvalues with maximum precision in Mathematica Dec 3 comment How to obtain Eigenvalues with maximum precision in Mathematica You beat me to it! +1 Nov 22 comment Why Mathematica (Reduce) can't find clear solution for almost trivial inequalities? @Tamas: The $b\neq0$ assumption is not included in the $b^2>4ac$ assumption, since $c$ can be less than zero. This is (almost) clear from the image posted in my answer. PS If you're happy with the answer, press the tick button to accept it! Nov 21 comment Why Mathematica (Reduce) can't find clear solution for almost trivial inequalities? @J.M.: Sure, for one off assumptions, the explicit argument or Assumptions option or Assuming[] is best. Assuming basically just Blocks $Assumptions anyway. Nov 21 comment Why Mathematica (Reduce) can't find clear solution for almost trivial inequalities? I just noticed that this is basically what J.M. did in the comments above (although the Backsubstitution -> True was not necessary). Nov 21 answered Why Mathematica (Reduce) can't find clear solution for almost trivial inequalities? Nov 21 comment Why have we chosen our number system to be decimal (base 10)? There's a Wikipedia article on radix economy that gives the argument for base$e$being the most efficient. Nov 20 revised Finding all vectors of$\vec{y}$such that$\operatorname{span} \left \{ \vec{v_1}, \vec{v_2}, \vec{y} \right \} = \mathbb{R^3}\$ added 101 characters in body