Let $A$ be a diagonalizable matrix over the field $F$ and $f:F\rightarrow F$. Then we can define the following matrix:

$f(A) = P^{-1}f(D)P$ where $D$ is diagonal and $f(D)$ is the diagonal matrix defined via $f(D)_{ii} = d(D_{ii})$

Alternatively we can define $f(A)$ via its eigenspaces $E_{x}(f(A)) = \sum\limits_{x\in f^{-1}(x)}E_{x}(A)$

Please assume $F$ is whatever we want if there is a special word for it in that context


As mentioned by Qiaochu Yuan in the comments, " This general construction is called “functional calculus,” see for example: https://en.m.wikipedia.org/wiki/Holomorphic_functional_calculus "


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.