If $A,B\in M(n,\mathbb{R})$ and there exists $P\in GL(n,\mathbb{C})$ such that $A=PBP^{-1}$, does that imply that there exists $Q\in GL(n,\mathbb{R})$ such that $A=QBQ^{-1}$?
121710
As in the title:  If two real (square) matrices are conjugated over $\mathbb{C}$, are they then also conjugated over $\mathbb{R}$?
5070
This was an assertion made in our textbook but I have no idea how to show that either statement is true.  Also would like to show that that $\mathbb{Q}(\sqrt{2})$ is strictly larger than $\mathbb{Q}$, which was the second part of the assertion.
96369
Is it true that if $x_1, \dots, x_n \in \Bbb C$ are algebraically independent over $\mathbb Q$, then they are algebraically independent over $\overline{\mathbb Q}$?  I know how to prove it for $n=1$, by the contrapositive. If $P(x)=0$ and $P \neq 0$ has coefficients $a_0,...,a_n$ in $\overline{\mathbb Q}$, then $x$ is algebraic over $\mathbb Q(a_0,...,a_n)$ which is itself algebraic over $\mathbb Q$, therefore $x$ is algebraic over $\mathbb Q$, so it satisfies some relation $Q(x)=0$ where $Q \neq 0$ is a polynomial with rational coefficients.  For $n \geq 2$, I tried induction, but I was not sure how to proceed.  Thank you for your help!
71386
I am fairly sure, given examples $\Bbb{R}\times \Bbb{R},\Bbb{R}\times \Bbb{Q},\Bbb{Q}\times \Bbb{Q} $, that this is correct, but do not know how to prove it.   In my cited examples the proof has always used some specific property of $A,B$, leaving me clueless as to the general method of solving this. I am especially interested in the case were $|\Bbb{R}|&gt;A&gt;|\Bbb{Q}|,|B|=|\Bbb{Q}|$, though ideally the answer would be more general.  To be explicit, I'm asking for a proof of the statement in the question title, where $A,B$ are sets with infinite cardinalities.
123308
I figured we'd have to show $a=qb$ from $2^a-1=r(2^b-1)$ for $q,r\in\mathbb{Z}$ for the first implication and vice versa, how should I proceed from here?
6879
Let $P=\{A \subseteq \mathbb N: A$ is non-empty, finite and has an even number of elements$\}$. Consider $X=\{A \in P : 1 \in A\}$. Does $X$ have a lower bound in $(P,\subseteq)$? I think that there is a lower bound.
67021
Suppose $f\in L^{p}(\mathbb{R}^{n}) \cap L^{q}(\mathbb{R}^{n})$. How can I prove that for any $p \lt r \lt q$,  $$ \lVert f \rVert_{r} \leq (\lVert f \rVert_{p})^{(1/r-1/q)/(1/p-1/q)} (\lVert f \rVert_{q})^{(1/r-1/q)/(1/p-1/q)}\:? $$ I tried using Hölder to show that $f$ is in $L^{r}$ but I'm completely lost on how to go about it...
23648
I am trying to prove a series of propositions:  Given any homomorphism p from $\mathbb{R}$[X] to $\mathbb{R}$[X], show that it is equal to $\phi_g$ for a unique g in $\mathbb{R}$[X], with $\phi_g$(f) = f(g(X)). I've expanded the expression $p(f) = p(\sum_{i=0}^n a_iX^i) =\sum_{i=0}^n p(a_i)p(X)^i$ but I'm not sure how to show $p(f) = \sum_{i=0}^n p(a_i)p(X)^i = \sum_{i=0}^n a_ig(X)^i = \phi_g(f)$.  Show that if h,g $\in \mathbb{R}$[X] are such that h(g(X)) = X, then g(X) = aX+b for a $\in$ R$^x$ and b$\in \mathbb{R}$.
73293
How to show that if $A, B \in M_{n}(\mathbb{C})$, and $x,y\in \mathbb{C}^{n}$ then the following two conditions are equivalent,   1.) $\langle Ax,y\rangle =\langle Bx,y\rangle$   2.) $\langle Ax,x \rangle= \langle Bx,x \rangle$   for all $x,y$?
108313
I'm trying to find a necessary and sufficient condition for $x^n - y^m$ to be irreducible in $\Bbb C[x,y]$.  What I have now is rather trivial sufficient conditions: if $m\mid n$ then $m=1$ and if $n\mid m$ then $n = 1$.  There are similar questions previously posted to this site. To me the answers to those questions seem overkill to this particular problem, and I am looking for an elementary solution.  I would be grateful for your help.
4932
Let $m &gt; n$ be positive integers. Show that there do not exist matrices $A ∈ R_{m×n}$ and $B ∈ R_{n×m}$ such that $AB = I_m$, where $I_m$ is the $m × m$ identity matrix  If someone could explain it with an example, it will be appreciable.
120562
Is there $n\in \mathbb{N}$ such that $\ln(\ln(n)) \in \mathbb{Q}$? If such $n$ exists, we will get $$\ln(\ln(n))  = \frac{p}{q}, \quad p, q \in \mathbb{Z}.$$  Hence we will get $n = e^{e^{p/q}},$ where the question about the nature of  $e^{e^{p/q}}$ haven't been answered yet.   Is there any other direction ?  Thank you in advance.
10631
Let $A\in M_3(\Bbb R)$ which is not a diagonal matrix. Let $P$ be a polynomial (in one variable), with real coefficients and of degree 3 such that $P(A) = 0$. Pick out the true statements:  a. $P = cP_A$ where $c \in\Bbb R$ and $P_A$ is the characteristic polynomial of $A$;  b. if $P$ has a complex root (i.e. a root with non-zero imaginary part), then $P = cP_A$, with $c$ and $P_A$ as above;  c. if $P$ has a complex root, then $A$ is diagonalizable over $\Bbb C$.  I guess b and c are true although I am unable to disprove a. Because A could satisfy $(X-A)$ but this is not a polynomial. Now if $P$ has a complex root, then its conjugate too is its root as the polynomial has  real coefficients. As we all know An odd degree polynomial has always a real root, what we have is three distinct roots. Since $A$ satisfies the polynomial the only difference between the characteristic polynomial and $P$ has to be a constant.
92415
We want to prove that a group, say $G$, of order $pqr$ where $p \gt q \gt r$ has a normal Sylow $p$-subgroup and deduce that it has a normal subgroup of order $pq$.    I know how to show $G$ has a normal Sylow $p$-subgroup or normal Sylow $q$-subgroup, but not the rest. Thanks for your advices.
7568
Let $A$ and $B$ be $m\times n$ matrices over $\mathbb{Z}$ such that image($A$) = image($B$), where $A$ and $B$ are considered as maps $\mathbb{Z}^n \rightarrow \mathbb{Z}^m$. Does there exist an invertible $n\times n$ matrix $P$ such that $A=BP$? what happens if we consider the matrices to be over $\mathbb{Q}$ instead of $\mathbb{Z}$?  I would prefer some hints instead of full answer.  Thanks in advance.
121387
Is it true or false that $\lvert P([0, 1]) \setminus P((0, 1)) \rvert &gt; \lvert P(\mathbb{N}) \rvert$ ?  I need a hint because I don't see a way to solve this
24224
I want to prove, that $GL(n,K)$ is open in $M(n,K)$, where ($K=\{\mathbb{R},\mathbb{C},\mathbb{H}\}$.  For the prove I don't want to use that the determinant is continous. Alternativly I assume a submultiplicative norm.  First step: Define $U_A:=\{B\in M(n,K):\Vert A-B\Vert&lt;\frac{1}{\Vert A^{-1}\Vert}\}$ and show that it is a neighbourhod of $A$ in $M(n,K)$.  Second step: I show that the series $\sum_{k\ge0}(1-A^{-1}B)^kA^{-1}$ converges absolutely and the limit is inverse to $B$.  For the second step I should use the Neumann series, right? But I have some problems with the first step. Any help?