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Accepted

### Is this proof of Perron's theorem correct, and if so is it original?

(1) Correctness: I read all arguments in detail and couldn't find anything wrong with them. Of course, this doesn't mean too much... (2) Orginality: I think in a topic which has such an extensive ...
Accepted

The formula you're looking for can be obtained by differentiating Jacobi's formula $$\frac{\mathrm{d}}{\mathrm{d}t} \det A(t) = \det A(t) \cdot \operatorname{tr}\left( A^{-1} \frac{\mathrm{d}A}{\... 35 votes Accepted ### Consequences of eigenvector-eigenvalue formula found by studying neutrinos The OP asks about generalisations and applications of the formula in arXiv:1908.03795. \bullet Concerning generalisations: I have found an older paper, from 1993, where it seems that the same ... 31 votes ### Proof for the derivative of the determinant of a matrix Another way to obtain the formula is to first consider the derivative of the determinant at the identity:$$ \frac{d}{dt} \det (I + t M) = \operatorname{tr} M. $$Next, one has$$ \begin{split} \...
Let $C := A^{1/2} (A+B) A^{1/2} + B^{1/2} (A+B) B^{1/2}$; this is a positive semi-definite matrix with the same trace as $(A+B)^2$. We show that the eigenvalues of $C$ are majorised by the ...