So I have this matrix I am working with that looks like $$S=\begin{pmatrix}0 & -a & b\\ a & 0 & -c\\ -b & c& 0 \end{pmatrix} $$ and upon asking mathematica for the eigenvalues I get Eigenvlaues[S] $=0, \sqrt{-a^2+-b^2+-c^2},-\sqrt{-a^2+-b^2+-c^2} $. Yes the eigenvalues are imaginary, but why doesn't it simplify it to be imaginary, and how can I force it to replace those minus sines and slap an $i$ in there instead? When I go to find the Eigenvectors does it recognize the terms I have as imaginary eigenvalues and simplify as such, or do I need to force it somehow.
$\begingroup$
$\endgroup$
6
a,b, andcare treated as complex numbers. -- Finally, something like this might work:PowerExpand[Sqrt[-a^2]]$\endgroup$ComplexExpand. $\endgroup$Simplify[Sqrt[-a^2 - b^2], a > 0 && b > 0],Simplify[Sqrt[-a^2 - b^2], a >= 0 && b >= 0],Simplify[Sqrt[-a^2 - b^2], {a, b} \[Element] Reals],Simplify[Sqrt[-a^2 - b^2], a != 0 && b != 0 && {a, b} \[Element] Reals]-- That the casea == b == 0seems important here looks like a weakness of Mathematica. It probably does not check whether the expression under the radical can change sign or not, probably for efficiency's sake. $\endgroup$0as a non-imaginary number and keep it separate from the (nonzero) imaginary numbers:Assuming[a^2 + b^2 >= 0, FullSimplify@PiecewiseExpand@ComplexExpand[Sqrt[-a^2 - b^2]] ]$\endgroup$