How to find the eigenvalues of the following tridiagonal matrix The $n$-order tridiagonal matrix is defined by $A(i,i)=0, A(i,i+1)=n-i, A(i+1,i)=i$, i.e., $$A=\begin{pmatrix}0&amp;n-1\\1&amp;0&amp;n-2\\&amp;2&amp;0&amp;n-3\\&amp;&amp;\ddots&amp;\ddots&amp;\ddots\\&amp;&amp;&amp;n-2&amp;0&amp;1\\&amp;&amp;&amp;&amp;n-1&amp;0\end{pmatrix}$$ All eigenvalues of $A$ are $-(n-1),-(n-3),\ldots,(n-3),(n-1)$, however, I could not figure out a simple way to show that.
Calculating the eigenvalues of a matrix How to find the eigenvalues of $$\begin{bmatrix} 0 &amp; 1   &amp;   &amp;  &amp;\\ k &amp; 0   &amp; 2 &amp;  &amp;\\   &amp; k-1 &amp; 0 &amp; 3 &amp;\\   &amp;     &amp;\ddots&amp;\ddots&amp;\ddots\\   &amp;     &amp;      &amp;2 &amp; 0 &amp;k \\   &amp;     &amp;      &amp; &amp; 1 &amp;0 \end{bmatrix}$$ I tried recurrence equation, but it doesn't work to find the characteristic polynomial. Any hint or solution are welcomed, thanks for your help!  PS:The missing parts are all zeroes.
This is kind of a weird request. For some reason whoever owns safeandbuy.com has pointed their domain at my IP address. The reason it's a problem is that I'm having all kinds of crawlers that are trying to crawl my site with that domain name.   Is there anything I can do about this? How to change programmatically exposed filters in views I have a page view with exposed filters. One of the filters is term vocabulary, and that's dropdown list, user can select only one category (term).   I want to set programmatically one term from dropdown list when I came from specific URL. I tried with views_query_alter where $category is tid.    $set_exposed_input  = array();  $set_exposed_input['field_category_tid'] = $category;   if ($set_exposed_input) {     $view-&gt;set_exposed_input($set_exposed_input);  } Comparison of backup tools This question exists because it has historical significance, but it is not considered a good, on-topic question for this site, so please do not use it as evidence that you can ask similar questions here.  While you are encouraged to help maintain its answers, please understand that "big list" questions are not generally allowed on Ask Ubuntu and will be closed per the .   Backup is incredibly important. Obviously there's no best backup tool, but a comparison of the options would be very interesting.   Graphical Interface? Command line? Incremental backups? Automatic backups? Install method: In standard repositories? PPA? Can someone tell me what's wrong with this? because i can't figure out where the missing \right( is.  \begin{equation}  \begin{split} Μ=&amp;-\pi \rho \frac{c^{2}}{4}    \left[c\left(\frac{α}{c}-\frac{1}{2}\right)\ddot{h}+Uc\left(\frac{3}{4}-     \frac{α}{c}\right)\dot{\alpha} +\frac{c^{2}}{4}\left(\frac{9}{8}+      \frac{4α^{2}}{c^{2}}-\frac{4α}{c}\right)\ddot{\alpha} \\  &amp;-\left(\frac{4α}{c}-1\right)U C(k) \left(-\dot{h} +U\alpha + c \left(\frac{3}{4}-\frac{α}{c}\right)\dot{\alpha}\right)\right]  \end{split} \end{equation} calculating expected number of packets. $S=\{(n,{1\over n}):n\in\mathbb{N}\}$ is closed in $X$? The topological space $X=\mathbb{N}\times \mathbb{Q}$ with subspace topology of $\mathbb{R}^2$ the set $S=\{(n,{1\over n}):n\in\mathbb{N}\}$ is closed in $X$?  $S^c=\phi\times\{{1\over n}\}^c$ right? but I know $\mathbb{Q}$ is neither open nor closed in $\mathbb{R}$, I am confused here, could any one help me to answer this question?  the boundary of $S$ is $\bar{S}\setminus $ int S. as int S is $\phi$ so boundary is $\mathbb{N}\times \mathbb{R}$? How does Newtonian mechanics explain why orbiting objects do not fall to the object they are orbiting? The force of gravity is constantly being applied to an orbiting object.  And therefore the object is constantly accelerating.  Why doesn't gravity eventually "win" over the object's momentum, like a force such as friction eventually slows down a car that runs out of gas?  I understand (I think) how relativity explains it, but how does Newtonian mechanics explain it? I would like to place a lower limit under two consecutive \sums, somewhere in between. So I have \sum_{j=1}^{\infty}\sum_{k=1}^{\infty} and would like to place j&gt;k in the lower limit between the two sums. I believe the multiple integral operator has this kind of functionality. Sign of Estimate (Coefficient) of Interaction Terms I create a linear regression model with interaction term in the model say: $$y=a_0+a_1x_1+a_2x_2+a_3x_1*x_2+e$$ where $x_1$ is continuous and $x_2$ is binary. Now, I have couple of questions:   If $x_1$ and $x_2$ both are significant and have positive coefficients, that means that if $x_1$ and $x_2$ increase, then $y$ increases. What can I say if $x_1*x_2$ is statistically insignificant? Does it mean that increase in. Does it mean that for $x_2=1$, the effect of $x_1$ on $y$ is not changed, i.e. increase in $x_1$ still leads to increase in $y$? What if the interaction is statistically significant? What can I say if the interaction term is statistically insignificant and negative? Does it indicate that if $x_2=1$, increase in $x_1$ actually leads to decrease in $y$?   Thanks. Variable-name \newcommand with parameters within another \newcommand I seem to have clicked on something that changed my display preferences. The main site is now very minimalistic, with fewer controls and different fonts.  I want to change it back, but where is it? I am looking for a way to clean up the mess when my top-level script exits.  Especially if I want to use set -e, I wish the background process would die when the script exits. Stars and pins are retained when chat messages are moved! In the chat system, stars and pins are retained when moved to another room. This is probably a bug. Here a picture:  When a room owner pins a message, they can move it to another room, and they cannot unpin it. However when a starred message is moved to another room, the user can unstar it in the room it was moved to. Conversation source:  * is irrelevant and does not have to do with this bug report. Many thanks to . There was this cool tv show I used to watch as a kid (in the 1990's I think). It was about a world where water was extremely rare and people used to fight over it, however there was this person who had this large 3D polyhedron shaped device (big enough to fit a few people in it) that could synthesise water.  Now, this device was either a time travel device or could just make water (not sure but I remember some time travel aspect as they could either remember the world the way it was where water was plentiful or could travel back to it on occasion).  Anyway, would love to know if this rings any bells?????  It was on in the UK (but maybe the US too, not sure). Examples and further results about the order of the product of two elements in a group Let $G$ be a group and let $a,b$ be two elements of $G$. What can we say about the order of their product $ab$?   says "not much":     There is no general formula relating the order of a product $ab$ to the orders of $a$ and $b$. In fact, it is possible that both $a$ and $b$ have finite order while $ab$ has infinite order, or that both $a$ and $b$ have infinite order while $ab$ has finite order.   On the other hand no examples are provided. $(\mathbb{Z},+), 1$ and $-1$ give an example of elements of infinite order with product of finite order. I can't think of any example of the other kind! So:   What's an example of a group $G$ and two elements $a,b$ both of finite order such that their product has infinite order?     Wikipedia then states:     If $ab = ba$, we can at least say that $\mathrm{ord}(ab)$ divides $\mathrm{lcm}(\mathrm{ord}(a), \mathrm{ord}(b))$   which is easy to prove, but not very effective. So:   What are some similar results about the order of a product, perhaps with some additional hypotheses? Easier way to debug a Windows service How do I loop through a MySQL query via PDO in PHP? I'm slowly moving all of my LAMP websites from mysql_ functions to PDO functions and I've hit my first brick wall.  I don't know how to loop through results with a parameter.  I am fine with the following:  foreach ($database-&gt;query("SELECT * FROM widgets") as $results) {    echo $results["widget_name"]; }   However if I want to do something like this:  foreach ($database-&gt;query("SELECT * FROM widgets WHERE something='something else'") as $results) {    echo $results["widget_name"]; }   Obviously the 'something else' will be dynamic. How to copy the complete folder structure one location to another location include all files and sub folders I have folder structure like in Shared documents/East using CSOM and C# Main Folder   Folder1   Folder2      File1      File2  I have to copy all folders with files to Shared documents/West .... Which characters are valid in CSS class names/selectors? Ok guys, continuing my passage through edwards... here is the question... thanks for hints/solutions in advance:  Suppose $f(x,y,z)=0$ can be solved for each of the three variables $x,y,z$ as a differentiable function of the other two. Then prove that  $\displaystyle \frac{\partial x}{\partial y} \frac{\partial y}{\partial z} \frac{\partial z}{\partial x} = -1$  Verify this is the case for the ideal gas equation $pv =RT$ where (where $p,v,T$ are the three variables and $R$ is the constant).