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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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    $\begingroup$ if $A$ is diagonal, I think $f$ is any function with a Taylor series $\endgroup$ – gt6989b Jul 20 '16 at 3:33
  • $\begingroup$ This sounds familiar, you must be able to find a lemma or something saying this. But there is an obvious observation: If $\lambda$ is an eigenvalue of $A$ with eigenvector $v$, then $A^k(v)=A^{k-1}(Av)=A^{k-1}(\lambda v)=A^{k-2}(A(\lambda v)=A^{k-2}(\lambda A(v))=A^{k-2}(\lambda ^2 v)=\cdots=\lambda ^k v.$ This shows that $\lambda ^k$ is eigenvalue of $A^k$. So, for $p(x)$ polynomial, it is true that if $\lambda$ is an eigenvalue for $A$, then $p(\lambda)$ is one for $p(A)$. By approximation, you may be able to extend this to many other functions, at least for those that have series expansion. $\endgroup$ – Behnam Esmayli Jul 20 '16 at 4:12
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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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  • $\begingroup$ So the radius of convergence of $f$ as a function from $\mathbb{R}\rightarrow \mathbb{R}$ determines the radius as long as the matrix is "small" under some norm? Is there a result or textbook I can refer to on this? Also, how does Cayle Hamilton give you your second line? $\endgroup$ – qbert Jul 20 '16 at 7:57

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