What does a function of matrices do to the eigenvalues of matrices in its domain? Two examples and request for generalization if possible
I think, for example, that if $\lambda$ is an eigenvalue of a matrix $A$, then $\lambda^2$ is an eigenvalue for $A^2$ and that $\frac{1}{\lambda}$ is an eigenvalue for $A^{-1}$ provided $A$ is invertible.
Is there a class of matrix valued functions functions $f$ for which the eigenvalues of $f(A)$ are $\widetilde{f}(\lambda)$, where $\widetilde{f}$ is the "analogous" scalar function? Not sure how to say the last part precisely which is also why it would be great if someone could direct me towards the terms to look up.
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$\begingroup$Given any real analytic function in a neighborhood of $0,$ we get $f(A)$ defined as long as some induced norm for $A$ is smaller than the radius of convergence for the Taylor series of $f.$ Or, of course, if the radius is infinite, as in $e^x.$
However, the Cayley Hamilton Theorem says that $f(A)$ can be rewritten as a polynomial in $A,$ of degree no larger than $n.$
You just need to figure out your question for polynomials of various types of Jordan normal forms.
Notice that this does not directly apply to $1/x,$ which gives $A^{-1}$ only if $A$ is actually invertible. However, it does apply to $1/(1+x),$ or $(I+A)^{-1}$ when $A$ is near the $0$ matrix.
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