Are there any general relations between the eigenvalues of a matrix $M$ and those of $M^2$?
Relationship between eigenvalues of a matrix and its square
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matrices
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1Roughly speaking, the eigenvalues of $f(M)$ are $f(\lambda)$, where the $\lambda$ are eigenvalues of $M$. In this case $f(x) = x^2$. – 2012-11-02
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2Not just "roughly": it's always the case (with eigenvalues counted by algebraic multiplicity) for any polynomial, or more generally (using the holomorphic functional calculus) for any function $f$ analytic in a neighbourhood of the set of eigenvalues. – 2012-11-02