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Mathematica and square roots

Dear All!
I am new to this group. May be this question was already discussed,
but I did not find solution, searching this group. I was calculating
eigenvalues {x} and eigenvectors {v} of a square matrix A. The
characteristic polynomial looks like
(x^2 - a^2) (...) = 0

Command Eigenvalues[A] gives me eigenvalues
x1 = +a; x2 = -a; x3 = ...
Command Eigenvectors[A] gives corresponding eigenvectors.

The problem is that 'a' in my computations can be both negative and
positive. So, eigenvalues x1 and x2 can be either negative or positive
depending on sign 'a'. It is much more convenient for further
computations to have eigenvalues of fixed sign. I want my eigenvalues
look like
x1 = +Abs[a]; x2 = -Abs[a]; x3 = ...
And, of course, I want to have corresponding eigenvectors.

Is it possible to do this using Eigenvalues and Eigenvectors, or I
have to solve equation
A v = Abs[a] v
"manually" to find eigenvector 'v' which corresponds to the eigenvalue

Thank you in advance.

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