Let $A\in L(V), A = \begin{bmatrix}0&amp;0&amp;...&amp; 0 &amp; \alpha_{1}\\0&amp;0&amp;...&amp; 0 &amp; \alpha_{2}\\\vdots&amp;\vdots&amp;\ddots&amp;\vdots&amp;\vdots\\0&amp;0&amp;...&amp; 0 &amp; \alpha_{n-1}\end{bmatrix}$. Determine its characteristic and minimal polynomial.   I have concluded that $k_{A}(\lambda) = (n-1- \lambda)(-\lambda)^{n-1}$, but I am stuck on minimal polynomial. My strategy was to observe $(A - \lambda I),(A - \lambda I)^{2},...$ , until I get matrix $(A - \lambda I)^{k}$ such that $\dim(Im ((A - \lambda I)^{k})) = n-1$, but since all those matrices are diagonal, my understanding is that we will never achieve that.
I have come across a question about determining possible minimal polynomials for a rank one linear operator and I am wondering if I am using the correct proof method.  I think that the facts needed to solve this problem come from the section on Nilpotent operators from Hoffman and Kunze's "Linear Algebra".     Question: Let $V$ be a vector space of dimension $n$ over the field $F$ and consider a linear operator $T$ on $V$ such that $\mathrm{rank}(T) = 1$.  List all possible minimal polynomials for $T$.   Sketch of Proof:   If $T$ is nilpotent then the minimal polynomial of $T$ is $x^k$ for some $k\leq n$.  So suppose $T$ is not nilpotent, then we can argue that $T$ is diagonalizable based on the fact that $T$ must have one nonzero eigenvalue otherwise it would be nilpotent (I am leaving details of the proof of diagonalization but it is the observation that the characteristic space of the nonzero eigenvalue is the range of T and has dimension $1$).  Thus the minimal polynomial of $T$ is just a linear term $x-c$.    Did I make a mistake assuming that $T$ can have only one nonzero eigenvalue?  Thanks for your help
How is The Wall bypassed? grep: memory exhausted Assume a p x n data matrix  (e.g. p independent samples of n variables). The n variables are correlated according to a correlation matrix $\rho$. Now if you consider the data matrix as some state t (t=time)  and based on this state I would like to calculate the next timestep (t+1). Therefore the correlation matrix would be allowed to have values &lt; 1 in the diagonal, which means that the diagonal elements would indicate the influence from each variable $x_{p,i}[t]$ on $x_{p,i}[t+1]$. So I would be interested to find a function f like this: $x_{p,i}[t+1] = f(x_{p,i}[t],\rho)$ so that the p x n data matrix  consisting of the $x_{p,i}[t+1]$ replicates the $\rho_{ij}$ (except the diagonal element). f should be something like this: $x_{i,j}[t+1] = \sum_{(k=1\ldots n)}\rho_{i,k} \cdot x_{kj}[t] + e_{i,j}.$ This means specifically, I not only want to calculate a data matrix, that replicates the correlation matrix as such, but also to let each observation be depending on its own state and linear dependent on the previous state of all the intercorrelated variables of its own state. For ease one can assume that the p variables are distributed normal and $e_{ij}$ are independent and distributed with mean$(e_{i,j}) = 0 $. Then for a numerical simulation it would be interesting  to find a formula to calculate the variance/standard deviation of each of the j variables (sd($e_{[.,j]} = ?).$ Thank you for any help! A customer has a 6-screen setup for each workstation and would like various pages of a web app to be opened automatically in each of the screens on startup or invoking a shortcut. (The target web browser is likely to be Chrome).  I wondered whether it would be possible to use Javascript window.open to do this but one possible snag is the customer may wish to have tabbed browsing with popup blocker turned on. (OK, these could be overridden but such config is currently out of our control...)  So the question is: Could a script (e.g. batch file) be written to open multiple instances of the browser at specified coordinates within the 6 screens (each pointing to different URLs)?  [This is basically the same question as  but for Windows instead of Ubuntu.] How can I modify the advanced search form when my hook runs too early? I'm using  to modify the content of the form on the search page &mdash; I want to remove the "phrase" and "not" fields, and restrict the "only of the types" checkboxes to a hand selected list of types.  I have the following code, which is being called and which I think should work:  function MYMODULE_form_search_form_alter(&amp;$form, &amp;$form_state, $form_id) {     unset($form['basic']);     unset($form['advanced']['keywords']['phrase']);     unset($form['advanced']['keywords']['negative']);     unset($form['advanced']['type']);     $form['advanced']['type'] = array(         '#type' =&gt; 'checkboxes',         '#title' =&gt; t('Only of the type(s)'),         '#options' =&gt; array(             'blog' =&gt; 'Blog entry',             'event' =&gt; 'Event',             'news' =&gt; 'News article'         )     );     // print_r($form);     // die; }   Except  gets to the search form after my form alter hook runs, so I don't get a chance to modify any of the form elements that it's adding.  I'm running Drupal 7.22.  How can I modify the advanced search form when my module hook runs before the hook that defines the advanced search form? Initialization of an ArrayList in one line I find myself frequently needing to rename different files with mv that are in deep directories:  mv /a/long/path/to/a/file.txt /a/long/path/to/a/file.txt.bak   But I don't want to retype the full path name. Is there a quick shorthand or alias that I could use? I.e.:  $ mv /a/long/path/to/a/file.txt file.txt.bak $ ls /a/long/path/to/a/file.txt.bak a/long/path/to/a/file.txt.bak   (note: this is for straightforward, single file renames in different directories at different times, not for ) So what I'm looking for here is something like PHP's  function.  This is so I can debug my scripts by seeing what's the state of the object in question. I've inherited a Maintenance Plans that does the following:   Cleanup old data Checks DB integrity Performs Database and Transaction Log Backups Reorganizes Our indexes Updates Statistics Delete old backups and Maintenance Plan files   Of the 23 minute Maintenance Plan, Updating the Statistics takes a staggering 13 minutes. During this 13 minute period, access to the database is blocked (or at least, replication from this DB to our others is paused).  My Question Is:  When should we be Updating the Statistics, and why?   This seems like the kind of thing we should do less frequently than every day. I'm trying to get us out of the "just because" mind-set of doing unnecessary Maintenance. How about a difficulty rating for questions? It might be useful to add a drop down field "Difficulty" for questions. With a very limited set of options, say:   Beginner Advanced Expert   This way, users could easily judge if they can answer a specific question (depending on their experience in the topic) and how much time it will probably take.  Beginners usually know they are new so they can easily select Beginner. Those with some experience just select Advanced and those working on a topic for years may chose Expert. I think it's not very difficult to judge halfway correctly.  And the difficulty rating does not have to be very precise anyway, it's more a rough estimate by the asking person.  As an addition, rep gained on this topic can be multiplied with a difficulty factor. This would solve the problem of lots of rep for often-viewed and easy to answer questions and few rep for less-viewed, difficult questions.  Update  Thank you so far for the answers. I clearly underestimated the fact that some users are going to always chose Expert difficulty.  Maybe readers voting on the difficulty would be a workaround. Of course, we cannot use the rep factor then, for the same reason.  This kind of voting would obviously introduce a whole new set of issues, since we could vote on both content and difficulty (I'm almost positive this kind of proposal existed before?). Can I shrink the transaction log file on a mirror database? This is a followup question to  about why I couldn't shrink the log file on the principal database.  To make a long story short, I setup database mirroring but forgot to make sure the job that backed up the transaction log was running again, and the transaction log grew to almost 60GB.   Since mirroring was set up, this size increase was duplicated on the mirrored server, and eventually took up all the disk space and made the mirror database unusable.  Per  about transaction log maintenance on a mirror database, you can't back up the log on the mirror, but when asked specifically in  about how to shrink an overgrown log file on a mirror database, a  was left that      One way to do this would be to failover to the mirrored database and   do the shrink there. Thoroughly test this out in a non-production   environment to ensure it has the behavior you want/expect.   This seems to suggest that there might be other ways to shrink the log file on a mirror, and this method might not necessarily be safe to do on a production server.  Is there a way to safely shrink the transaction log file on a database mirror? How to encrypt external devices using a cross-platform solution? What is the advantage of using sudo apt-get autoremove over a cleaner app? Workflow for converting LaTeX into Open Office / MS Word Format I often have to write up reports based on the analysis of some data. I use R to analyse the data and export tables, figures, and text. This is then included into a LaTeX document either using input or Sweave ().  However, when I collaborate with others, I sometimes need to provide a document in Open Office / MS Word format.  Question:  Thus, assume the simplest scenario   I have a LaTeX document with text, tables, and figures I need to export this reliably into Open Office or MS Word format: this includes mathematical formulas, table formatting, and quality figures) I don't need to go back from MS Word to LaTeX   What is a reliable, efficient, and preferably free process?  Initial Thoughts:   Export to HTML,  LaTeX - LyX - ODF Use odfWeave  that deals with some specific issues associated with LaTeX to Open Office conversion    I was hoping that there is an expert out there who has worked out a good system already. Say I have an embedded device which is only capable of doing HMAC-SHA1 transformations (that device is, in fact, a YubiKey).  Would it be secure to feed it like a block cipher in counter mode to get a random pad suitable as an stream cipher?  Of course, I'm talking about not repeating the counter value ever to encrypt data (so this construction would also need some random IV or treating the counter value as a nonce forever). Show that $S$ is compact. Let $A \in M_n(\bf C)$ and let   $S=\{ A \in M_n(\bf C) : A=A ^*$ and  $\rho (A) \leq 1 \}$, $\rho(A)$ denotes spectral radius of $A$.     Show that $S$ is compact subset of $M_n(\bf C) \cong \bf C^{n^2} $ with usual topology.    $S$ is subset of $\bf O_n(C)$ which is compact so I think its enough to show that $S$ is closed.Please help. What algorithm is used by computers to calculate logarithms? When one should use spacing line \quad or \, How to get time difference in minutes in PHP How to calculate minute difference between two date-times in PHP? When to use “that” and when to use “which”, especially in relative clauses