It is defined for numerical values only. Your have to assign numerical values to your symbols as variables, so it will work. Symbols or variables could be computed by supplying numerical values (it play similar as electric voltage or soul in the body ).

Of course it's possible-that it takes a long time is computer-dependent, since the roots of any fourth degree polynomial can be computed.

So defining, for example

A=Table[a[i,j],{i,1,4},{j,1,4}]

Eigenvalues[A]

does give an immediate answer, though it does then require considerable postprocessing, to make it useful.

You can find a symbolic solution of algebraic equation

Solve[x^4+a x^3+b x^2+c x+d==0,x], then obtain coefficients of the characteristic polinomial y=Series[Det[A - x*IdentityMatrix[4]],{x,0,4}] to replace a,b,c,d by values

a=SeriesCoefficient[y,{x, 0, 3}];

b=SeriesCoefficient[y,{x, 0, 2}];

c=SeriesCoefficient[y,{x, 0, 1}];

d=SeriesCoefficient[y,{x, 0, 0}];

Dear Afonso Dimas Martins

In addition to the helpful answers, I have a little bit to add.

Your question:

Is it possible to find the eigenvalues of a 4x4 symbolic matrix?

The answer is yes.

We need to apply the famous formulas of Ferarri or Cardano method

to solve the fourth-order characteristic polynomial of the 4x4 matrix.

In general, the final answer is not practical where it includes 16 symbols

in concentric radicals.

see

Quartic function - Wikipedia

en.wikipedia.org › wiki › Quartic_function

By assigning numeric for practicability, its computation is simple as : 4 x 4 case: https://www.youtube.com/watch?v=kK-Uj2nJgus

3 x 3 case: https://au.mathworks.com/help/symbolic/create-symbolic-matrices.html

Yes, we can find. Though it's time taking process to find the determinant of 4x4 matrix, but the shortcut method is that, you just expand the determinant in that row or column in which the maximum number of zeros are there, then you can find out the eigen value.

It's very possible

I would add that 4 is the highest matrix order when it is possible to find eigenvalues of symbolic matrix in radicals (Abel–Ruffini theorem): https://en.wikipedia.org/wiki/Abel%E2%80%93Ruffini_theorem