How should I get symbolic expression of eigenvalues and eigenvectors of a real symmetric matrix 3x3

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You can access the documentation for eigenvalues_right:

sage: gamma.eigenvectors_right?

or the source code:

sage: gamma.eigenvectors_right??

There you see that the right eigenvectors are computed by computing the left eigenvectors of the transpose matrix. So let's check the documentation for eigenvectors_left:

sage: gamma.eigenvectors_left?

or the source code:

sage: gamma.eigenvectors_left??

You can see there an example involving a 2x2 matrix with symbolic entries, followed by the following comment:

This routine calls Maxima and can struggle with even small matrices
with a few variables, such as a `3\times 3` matrix with three variables.
However, if the entries are integers or rationals it can produce exact
values in a reasonable time.

Note: to get the documentation for that command in html format, do:

sage: browse_sage_doc(gamma.eigenvectors_left)