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As we know, the eigen vectors and eigen values of a real symmetric matrix always is are real numbers. I am trying to use the Mathematica to verify this theory. Suppose I have a matrix A

A={{6585, 7579, 6717}, {7579, 11002, 12324}, {6717, 12324, 17030}}
Eigenvalues[A]

{Root[-13826467396+117514474 #1-34617 #1^2+#1^3&,3],Root[-13826467396+117514474 #1-34617 #1^2+#1^3&,2],Root[-13826467396+117514474 #1-34617 #1^2+#1^3&,1]}

But I really don't like those Root, so I try to simplfy it to a radical expression.

Simplify /@ ToRadicals /@ Eigenvalues[A]

The result includes I. Since they are real numbers, could we express this number by radical express without I?

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No, you cannot generally express roots of a polynomial of degree higher than two using radicals without using I. However, Root objects in Mathematica are numbers (they just can't be expressed rationally). In particular, for your problem:

Im[Eigenvalues[A]]
(* {0, 0, 0} *)
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As Carl Woll already said in comments, it is in general not possible to express the result in terms of radicals. However one can get a real result in terms of trigonometric functions.

You can use ComplexExpand to get a manifestly real expression:

 ComplexExpand[ToRadicals /@ Eigenvalues[A]]
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