Easy Gerschgorin Circle Theorem question?

If I have a matrix with complex entries, does the Gerschgorin Circle Theorem still hold?

If yes, the approximations for the eigenvalues will not be complex (since when calculating the radius we take the absolute value of the off-diagonal entry), but the eigenvalues of the actual matrix are actually complex?

So is it true that we can have our eigenvalue approximations not being complex even though the actual eigenvalues are complex?

Or equivalently, do we take the absolute of the element if it’s complex?


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