Eigenvalues of a symbolic 28×28 Hessian matrix

I have a little problem trying to calculate the eigenvalues of big symbolic (depending on $a$) Hessian matrices (28×28, 32×32 ...).

I think I've understand, by reading other posts, that Mathematica can not calculate the eigenvalues of such matrices since, I don't know why, it can not find the roots of the characteristic polynomial.

My goal is to find the lowest eigenvalue to see at which $a$ it vanishes. How could I do that since I don't have any idea at which $a$ it happens ? Thanks you

• is it possible to provide the smallest available instance of the problem? – user42582 Aug 27 '17 at 15:05
• If the goal is to find when an eigenvalue vanishes, why don't you look at the determinant instead? – bill s Aug 27 '17 at 16:30
• Well actually that a good solution, i'm pretty dumb since if H is my matrix and a_1,a_2,...,a_q are its eigenvalues Det[H]=a_1*...*a_q right ? – jack gredart Aug 27 '17 at 18:41
• Oh no i remember now, I already tried to calculate the Det,but it mathematica gives me something like : (1/((3.33262*10^-261 - 4.58187*10^-262a)^5)) 5.760357996978126*10^1833 a^7 (0.*10^-3127 + 0.*10^-3127 a + 0.*10^-3128 a^2 + 0.*10^-3128 a^3 + 0.*10^-3129 a^4 + 0.*10^-3129 a^5 + 0.*10^-3130 a^6 + 0.*10^-3131 a^7 + 0.*10^-3132 a^8 + 0.*10^-3134 a^9 + 0.*10^-3135 a^10 + 0.*10^-3137 a^11 + 0.*10^-3139 a^12) – jack gredart Aug 27 '17 at 18:46
• Since most of your constants in your Det are 0.* 10^-3131 or smaller it looks like you have some decimal numbers in your matrix which don't have more than 3130 good known digits after the decimal point. Try redoing your Det with say 6260 digits of precision for every constant and see what you get. Otherwise your N[det]==0. (to one digit of precision) – Bill Aug 27 '17 at 21:01