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This question already has an answer here:

I have read common pitfalls page and the documentation page but I still don't understand what do those signs mean here?

A = {{0.472, 0.008 + 0.01 I, 0.056 - 0.029 I}, {0.008 - 0.01 I, 0.235,
     0.003 - 0.002 I}, {0.056 + 0.029 I, 0.003 + 0.002 I, 0.293}};
Cvol = {{1, 0, 1/3}, {0, 2/3, 0}, {1/3, 0, 1}};
Eigenvalues[A - a*Cvol]
(*{1. Root[(-0.0531814 - 
       2.9302*10^-20 I) + (0.439425 - 
        6.61797*10^-19 I) a - (1.17112 + 0. I) a^2 + 
     1. a^3 + ((0.529735 - 3.65918*10^-19 I) - (2.88169 + 0. I) a + 
        3.75 a^2) #1 + (-1.6875 + 4.5 a) #1^2 + 1.6875 #1^3 &, 1], 
 1. Root[(-0.0531814 - 
       2.9302*10^-20 I) + (0.439425 - 
        6.61797*10^-19 I) a - (1.17112 + 0. I) a^2 + 
     1. a^3 + ((0.529735 - 3.65918*10^-19 I) - (2.88169 + 0. I) a + 
        3.75 a^2) #1 + (-1.6875 + 4.5 a) #1^2 + 1.6875 #1^3 &, 2], 
 1. Root[(-0.0531814 - 
       2.9302*10^-20 I) + (0.439425 - 
        6.61797*10^-19 I) a - (1.17112 + 0. I) a^2 + 
     1. a^3 + ((0.529735 - 3.65918*10^-19 I) - (2.88169 + 0. I) a + 
        3.75 a^2) #1 + (-1.6875 + 4.5 a) #1^2 + 1.6875 #1^3 &, 3]}*)
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marked as duplicate by Michael E2, MarcoB, Cassini, Mr.Wizard Feb 6 '16 at 17:48

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The solution to your Eigenvalues function are Root objects, see the documentation.

Root[f,k] represents the exact k^(th) root of the polynomial equation f[x]==0.

Inside Root is the pure function in the argument #1. If we named this pure function f with argument x, this would correspond to the situation in the above quote with f[x].

Your Matrix $A$ is from $\mathbb{C}^{3\times 3}$, so the eigenvalues are solutions of the characteristic polynomial of order $3$. The #1, according to the documentation

represents the first argument supplied to a pure function.

Because of the parameter a mathematica returns an abstract Root object instead of a closed solution.

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