Suppose that $Tr(A^n) = 0$ for all integer $n&gt;0$.  I have to prove that $A$ is nilpotent.  I can see it's true for little $n$ but i do not know how to prove that for all integer $n$. Any idea please?
$\DeclareMathOperator{\tr}{tr}$Let's consider a $n \times n$ matrix and the sequence of traces $\tr(A)=\tr(A^{2})= \ldots = \tr(A^{n})=0$. How to prove that $A$ is a nilpotent matrix (a matrix so that $A^{k} \times u = 0$ for all $u \in V$ and for some $k$)?  Maybe it would be reasonable to consider $A$'s Jordan form (we assume that the action occurs over an algebraically closed field, such as $\mathbb{C}$). For example, suppose that we have found it and $A'$ consists of $m$ blocks of sizes $r_{1}, \ldots r_{m}$ ($r_{1}+r_{2}+\ldots+r_{m}=m, r_{i} \in \mathbb{Z}_{+}$) with $\alpha_{1}, \ldots \alpha_{m}$ as corresponding eigenvalues to each of the blocks. Then, according to the problem will get the system  $$\begin{align} r_{1}\alpha_{1}+&amp;\ldots+r_{m}\alpha_{m}=0 \\ r_{1}{\alpha_{1}}^{2}+&amp;\ldots+r_{m} {\alpha_{m}} ^{2}=0\\ &amp;\ ~\,\vdots \\ r_{1}{\alpha_{1}}^{n}+&amp;\ldots +r_{m}{\alpha_{m}}^{n}=0  \end{align}$$  But the system's analysis over $\mathbb{C}$ seems to be too much complicated.  Actually, the problem must have more or less simple solution.  Any piece of advice would be much appreciated.
Shell script to delete directories older than n days Sending multiple messages through only one SocketChannel How to Send Byte array To Server glob() - sort array of files by last modified datetime stamp I'm trying to display an array of files in order of date (last modified).  I have done this buy looping through the array and sorting it into another array, but is there an easier (more efficient) way to do this? mysql database no space available In Chiswell's Mathematical Logic, there is the following problem is posed using the language of propositions (LP):     In some systems of logic (mostly constructive systems where 'true' is taken to mean 'provable') there is a rule      if $(\Gamma\vdash(\phi\,\lor\psi))$ is a correct sequent then at least   one of $(\Gamma\vdash\phi)$ and $(\Gamma\vdash\psi)$ is also correct.      By giving a counterexample to a particular instance, show that this is   unacceptable as a rule for LP. [Start by giving counterexamples for   both the sequents $(\vdash p_0)$ and $(\vdash(\neg p_0))$.]   In my , it was answered that one can let $\Gamma = \{p_0\,\lor\,(\neg p_0)\}$.  From there, one can deduce neither $(\Gamma\vdash p_0)$ nor $(\Gamma\vdash (\neg p_0))$.  But this does not proceed by the counterexample route that the author suggests.  Namely, he seems to imply that we should (1) take an interpretation of propositional variables $p_0,\,p_1,\,\ldots$ (setting each one equal to T or F), and (2) write $\phi$ and $\psi$ as functions of these propositional variables, and (3) pick sentences for $\Gamma$ which are all T, all with the conditions that (A) $(\phi\,\lor\psi)$ is evaluated as T, yet (B) both $\phi$ and $\psi$ are evaluated as F.  This, of course, seems impossible.  What, then, am I missing?  It seems that taking a concrete interpretation of the propositional symbols makes the constructivist's rule irrefutable.  (Note: this is supposed to be a very simple problem set, and I am an elementary logician). If $f$ is integrable then $|f|$ is also integrable Show that if $f$ is integrable on $[a, b]$ then $\lvert f \rvert$ is also integrable. Hint: Show that $$U (P , \lvert f \rvert) − L(P , \lvert f \rvert) ≤ U (P , f ) − L(P , f ).$$ I have: $$U (P , \lvert f \rvert) \ge U (P , f )$$ and $$ L(P , \lvert f \rvert) \ge L(P,f)$$ Thus, $$U (P , \lvert f \rvert) − L(P , \lvert f \rvert) ≤ U (P , f ) − L(P , f ).$$ How to prove $\sum_{k=1}^{n}F_k = F_{n+2}-1$ by induction when $F_n$ is the Fibonacci sequence Let $F_n$ be the Fibonacci sequence where  $F_0$ = 0  , $F_1$ = 1 and  $F_n$ = $F_{n-1}$ + $F_{n-2}$.                   I want to prove the following by induction. $$\sum_{k=1}^{n}F_k = F_{n+2}-1$$ Here is what I have so far.  Can anybody tell me if I am right?  Base case = n=1  $F_k$ = 1 = LHS  $F_{1+2}$ - 1 = 2 - 1 = 1 = RHS  Statement is true for n=1.  Assume statement is true for n=i.  $$\sum_{k=1}^{i}F_k = F_{i+2}-1$$  Prove statement is true for n=i+1.  $$\sum_{k=1}^{i+1}F_{k} = F_{((i+1)+2)}-1$$  $F_1$ + $F_2$ +...+ $F_i$ + $F_{i+1}$ = $F_{((i+1)+2)}-1$  $F_{i+2}$ - 1 + $F_{i+1}$ = $F_{((i+1)+2)}-1$  $F_{((i+1)+2)}-1$ = $F_{((i+1)+2)}-1$  Statement holds for n=i+1 Android EditText listener for cursor position change /usr/bin vs /usr/local/bin on Linux How could it possible to factorise $x^8-1$ in product of irreducibles in the rings $(\mathbb{Z}/2\mathbb{Z})[x]$ and $(\mathbb{Z}/3\mathbb{Z})[x]$? How could it possible to factorize $x^8-1$ in product of irreducibles in the rings $(\mathbb{Z}/2\mathbb{Z})[x]$ and $(\mathbb{Z}/3\mathbb{Z})[x]$?  I'm having a hard time starting the problem. Could anyone help me at this point? How many possible electromagnetic wavelengths are possible? Disclaimer: Please keep in mind that I am a young highschool student with no background in physics, this research was done in the course of an hour, and my reasoning could very well be wrong.  After a short conversation about color, I did some thinking, and read  The answer claims that there are $\infty^{\infty}$ colors   My first problem is the definition of colors, I'm going to simplify this (although it's not the same thing) as electromagnetic frequencies. This is what I mean when I refer to color in the rest of this question  The problem with $\infty^{\infty}$ colors is that the plank length exists. This causes two problems to arise with $\infty^{\infty}$. If $l_p$ is assumed to exist (as it is), this means the smallest distance in the universe is $l_p$. So, all wavelengths of light can be divided into a set, of the wavelengths produced by objects at $0^{\circ}$ Kelvin to $\infty^{\circ}$ Kelvin. According to this logic, there are simply $\infty$ colors.  Things are complicated further, when you introduce the Plank Temperature ($P_t$). $P_t$ is the theoretical limit temperature of the universe (at least for the purposes of light and color) Because an object hotter than $P_t$ will produce light with wavelengths shorter than $l_p$. This would introduce an upper limit on colors too.  Wouldn't this mean that the total number of possible colors is equal to however many plank lengths difference there is between an electromagnetic wave produced by a 0 degree object and a $l_p$ object? And therefor far less than $\infty$? Does this reasoning make sense at all? Horn and spindle tori Quantum Field Theory without LSZ, how is it possible? How many objects are being created? how do we know the BBP formula for $\pi$ is valid? I recently read about the Bailey–Borwein–Plouffe formula for calculating the $n^{\rm th}$ digit of $\pi$. I'm curious to how can we be sure that the formula is always accurate or correct?! Even if we can verify for every $\pi$ digit discovered till now with a "standard formula", how do we know that the formula is correct after those digits?  Also, what's the implication of these kind of BBP formulas for other "divine"/fundamental constants? can we learn anything else about the value of such constants and our universe through these? Any insight into the "supreme fascist" mind? I understand what a decibel is, the spirit of the question is different: why do we still use decibels (and with at least two different, context-specific definitions), if we have the feat of scientific notation? I dare to claim that the change of 5e-1 (which is a half) is much more readable than 3dB (voltage? power?). 1e6 is more clear than 120dB, etc. Just for the joy of adding? 3dB + 7dB is 10dB, but 2e5+3e6 is errrm... I see many new datasheets from the top vendors (LTC, for example) use the V/V convention, which is closer to the scientific notation than to the decibels. An upcoming paradigm change? I just got off a very long and very frustrating phone call with Apple support.  I purchased a used Apple computer which came installed with Mountain Lion. Meaning that the computer was shipped by Apple with Mountain Lion installed, and received by me with that same installation.  As a precaution, I tried to reinstall Mountain Lion via Recovery mode. This failed because my Apple ID hasn't purchased Mountain Lion.  Apple insisted that because I didn't purchase the computer new, I don't have a license for OS X. I don't believe this is correct, or even legal. This means that anyone using a used computer is doing so illegally, unless they've purchased a separate OS license.  Am I way off-base here? I'm pretty sure I now "own" a copy of Mountain Lion, and Apple is in the wrong. If this is correct, what are my options? In 1D why are vectors, such as, e.g., velocity $v$ and acceleration $a$, not written with a vector arrow? In my textbook instantaneous velocity and acceleration are wrote only as $v$ and $a$ in the chapter of One dimensional motion. Instead, in two dimensional motion chapter they are written with the "arrow" symbol upon them, $\vec{v}$ and $\vec{a}$.  Why if they are both vectors they don't have an arrow in that particular chapter? I just can't get it. Proof of Heron's Formula for the area of a triangle Let $a,b,c$ be the lengths of the sides of a triangle. The area is given by Heron's formula: $$A = \sqrt{p(p-a)(p-b)(p-c)},$$ where $p$ is half the perimeter, or $p=\frac{a+b+c}{2}$.  Could you please provide the proof of this formula? Thank you in advance.