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8h
comment Suppose $u,v \in \mathbb{C}$ are in the open unit disk. Is $|u|^n - |v|^n \leq |u - v|^n$?
Thanks for this. I will have to do my proof another way.
8h
comment Suppose $u,v \in \mathbb{C}$ are in the open unit disk. Is $|u|^n - |v|^n \leq |u - v|^n$?
Thank you for pointing out this easy counter-example.
8h
accepted Suppose $u,v \in \mathbb{C}$ are in the open unit disk. Is $|u|^n - |v|^n \leq |u - v|^n$?
9h
asked Suppose $u,v \in \mathbb{C}$ are in the open unit disk. Is $|u|^n - |v|^n \leq |u - v|^n$?
2d
comment What is the Laurent series of $\frac{2}{z-1} - z$ in $1<|z|<2$?
@C.Dubussy I see it now. So is it true then that this Laurent series converges as long as $1 < |z| <\infty$ holds? Hence, it converges within the upper bound of the annulus in the question.
2d
awarded  Mortarboard
2d
asked What is the Laurent series of $\frac{2}{z-1} - z$ in $1<|z|<2$?
Apr
23
revised Let $A$ be an $n \times n$ matrix over $\mathbb{C}$ or $\mathbb{R}$. Does $\det(e^A) = e^{\mathrm{tr}(A)}$ always hold?
deleted 3 characters in body; edited title
Apr
23
asked Let $A$ be an $n \times n$ matrix over $\mathbb{C}$ or $\mathbb{R}$. Does $\det(e^A) = e^{\mathrm{tr}(A)}$ always hold?
Apr
18
comment Is there a diagonalizable matrix $A \neq B$ such that $e^A = e^B$?
Helpful. This could also be generalized to any finite dimensional space by taking any diagonal matrix with diagonal entries $2 \pi i$, I believe.
Apr
18
asked Is there a diagonalizable matrix $A \neq B$ such that $e^A = e^B$?
Apr
16
accepted Let $z=x+iy$. What is $\frac{\partial}{\partial x} \log|z-w|$ and $\frac{\partial}{\partial y} \log|z-w|$?
Apr
16
asked Let $z=x+iy$. What is $\frac{\partial}{\partial x} \log|z-w|$ and $\frac{\partial}{\partial y} \log|z-w|$?
Apr
4
comment All solutions of $A^2+I=0$ in $M(n,\mathbb{C})$ are similar
How did you deduce the minimum polynomial from $A^2+I=0$? Is this a variant of Cayley-Hamilton? I calculated it as $p_A(x) = (i-x)^m (-i-x)^p$ which obviously does have repeated roots.
Apr
4
accepted All solutions of $A^2+I=0$ in $M(n,\mathbb{C})$ are similar
Apr
4
comment All solutions of $A^2+I=0$ in $M(n,\mathbb{C})$ are similar
In practice, we should be able to deduce the "appropriate" $m$ and $n$ from the characteristic polynomial of $A$, right?
Apr
4
comment All solutions of $A^2+I=0$ in $M(n,\mathbb{C})$ are similar
Shouldn't we mention that $A$ is diagonalizable over $\mathbb{C}$? I believe the diagonalizability of $A$ is why there are no $1$'s on the super-diagonal over the Jordan canonical form.
Apr
4
accepted Let $V$ be the space of complex polynomials on $[0,1]$. Is the differentiation operator self-adjoint?
Apr
3
asked When does the the eigenspace $E_\lambda(T)$ equal the generalized eigenspace $M_\lambda(T)$?
Apr
3
asked All solutions of $A^2+I=0$ in $M(n,\mathbb{C})$ are similar