Show that for every $A\in M_{n}(\mathbb{R})$ with rank(A)=1 that $||A||_{op}=||A||_{HS}$. What happens in the general case?     Just to clarify the notations:  $||A||_{op}= \sup\left\{\frac{\|Av\|}{\|v\|} : v\in V \mbox{ with &gt; }v\ne 0\right\}.$ $\|A\|^2_{HS}={\rm Tr} (A^{T}A) := \sum_{i \in I} \|Ae_i\|^2$   So using this inequality that I proved before:   $\|AB\|_{HS} \leq \|A\|_{\mathrm{op}} \|B\|_{HS} $ If I Substitute B for I I get the weak inequality, but I can't see how rank=1 helps me here..  Any help?
Problem:  Let $A_{n.n}$ be square complex matrix. Prove the following: $$\left \| A \right \|=\left \| A \right \|_{HS}\Leftrightarrow  rank(A)\leqslant 1$$. Where $\left \| . \right \|_{HS} $ is the Hilbert Schmidt Norm.  Please read my solution and tell me whether it is correct. If not, let me know where the mistake is.  Proof of the implication $\Leftarrow $:   If $rank(A)=0$, then in this case $A=0$. It follows that $\left \| A \right \|=\left \| A \right \|_{HS}=0$ If $rank(A)=1$, than: $ A=\begin{pmatrix} A_{1}\\  \alpha _{2}A_{1}\\  ...\\  \alpha _{n}A_{1} \end{pmatrix}$ where $A_{1}$ is the first row of $A$   and in this case: $\left \| A \right \|=\left \| A \right \|_{HS}=\left \| A_{1} \right \|\sqrt{1+\alpha _{1}^{2}+...+\alpha _{n}^{2}}$  Proof of the implication $\Rightarrow $   We know that $\left \| A \right \|=max_{\left \| x \right \|=1}\left \| Ax \right \|$  On the other hand: $ A=\begin{pmatrix} A_{1}\\  A_{2}\\  ... \\ A_{n}  \end{pmatrix}$. So, $$\left \| Ax \right \|=\left \| \begin{pmatrix} \left \langle A_{1},x \right \rangle\\  \left \langle A_{2},x \right \rangle\\  ...\\   \left \langle A_{n},x \right \rangle\end{pmatrix} \right \|=\sqrt{\left \langle A_{1},x \right \rangle^{2}+\left \langle A_{2},x \right \rangle^{2}+...+\left \langle A_{n},x \right \rangle^{2}} $$  Where $ \left \langle A_{i},x \right \rangle$ is the inner product of $A_{i}$ and $x$  Using the Cauchy-Schwarz inequality: $\left \langle A_{i},x \right \rangle^{2}\leq \left \| A_{i} \right \|^{2}\left \| x \right \|^{2}$, we get: $\left \| A \right \|=max_{\left \| x \right \|=1}\left \| Ax \right \|\leq max_{\left \| x \right \|=1}\left \| x \right \|\left \| A \right \|_{HS}=\left \| A \right \|_{HS}$. In order to have the equality: $\left \| A \right \|=\left \| A \right \|_{HS}$, we should have: $\left \langle A_{i},x \right \rangle^{2}=\left \| A_{i} \right \|^{2}\left \| x \right \|^{2} $ which occurs only if $A_{i}=\lambda _{i} x$. So, the rows of A are dependent, which implies that $rank(A)=1$. Note that the case $rank(A)=0$ happens when $A=0$  Please let me know if my solution makes sense.
Lightning aura:iteration over two objects with dynamic key I want to do simple thing - I want to iterate over list of two objects and use displayColumns as dynamic keys to get values from selectList object:  &lt;aura:attribute name="displayColumns" type="String[]"/&gt; &lt;aura:attribute name="selectList" type="Object[]"/&gt;   &lt;table&gt;     &lt;aura:iteration items="{!v.selectList}" var="elem"&gt;     &lt;tr&gt;         &lt;aura:iteration items="{!v.displayColumns}" var="col"&gt;             &lt;td&gt;{!elem[col]}&lt;/td&gt;         &lt;/aura:iteration&gt;     &lt;/tr&gt;     &lt;/aura:iteration&gt; &lt;/table&gt;   In pure JS I would do: elem[col.value], but in lightning compiler returns:      expecting a positive integer, found 'col' at column 6 of expression:   elem[col.value]: Source   Is there any workaround without creating proper object structure in controller? With the advent of the WordPress Answers Stack Exchange site, there's undoubtedly going to be overlap.  But I just want to get some advice:  I'm a developer and I kinda hate WordPress, but have to use it occasionally. My WordPress questions are usually development ones (when I have them), and most often related to 'how WordPress handles things', which are pretty obvious in straight programming terms.  But with the WordPress layer of abstraction, they become problematic.  I find that most developers that have gone near WordPress are usually also PHP/JS/CSS/MYSQL guys who dabble with WordPress and, as a result, hangout more on SO. And theres a die hard bunch who are heavy into the WP.SE site, but generally the WP site is for people trying to muddle through the day to day operation of using WordPress and its many themes/plugin etc.    As a result, if I were to ask something pretty in-depth about writing / integrating new &amp; strange features or something, I'm talking to a much smaller group of devs than I would be on SO.  So, enough of the ramble, my question:  Should I be posting heavy development questions relating to WordPress on the WP.SE site or to SO? Can I post to both, or is cross posting frowned upon in these cases still? The 1st law of thermodynamics If a gas expands adiabatically will the work done be positive or negative?  I think it will be positive as $\delta W=p dV$ and we have a positive sign due to the work done by the gas is positive as it pushes to expand. But this site has the negative of the answer I obtain.   (See adiabatic processes section) I am trying to prove the following by Mathematical Induction:  $$\sum_{i=1}^n i\cdot i! = (n+1)!-1\quad\text{for all integers $n\ge 1$}$$   My proof by Induction follows: First prove $P(1)$ is true,  $$\sum_{i=1}^1 i\cdot i! = (1+1)! - 1$$  Then, for all integers $k &gt;= 1$, if $P(k)$ is true then $P(k+1)$ is also true,  $$ \sum_{i=1}^k i\cdot i! = (k+1)! - 1$$ Then, we must show that $P(k+1)$ is true   $$ \sum_{i=1}^{k+1} i\cdot i! = ((k+1)+1)! - 1\\ \sum_{i=1}^{k+1} i\cdot i! = (k+2)! - 1$$  I am currently struggling with the next step to show that $P(k)$ is equal to $P(k+1)$. How would I finish this prove by induction? Using Newton's method calculate $ \frac{1}{\sqrt{a}} $ without division Suggest algorithm for the numerical calculation $ \frac{1}{\sqrt{a}} \ a &gt; 0 $ without division, use Newton's method.  My idea is:  $$ \frac{1}{\sqrt{a}} = (\sqrt{a})^{-1} = a^{-\frac{1}{2}}$$ $$ x = a^{-\frac{1}{2}} \Rightarrow f(x)=x^2 - a^{-1}$$ And next step is calulate $ f'(x)=2x $, but after substitution I got: $$ x_{n+1} = \frac{x_n^2+a^{-1}}{2x_n}$$ And know I still have divion, I guess. Maybe you've better idea on this task? Is Canonical Livepatch not supported on non-LTS release like Ubuntu 18.10? Last night I thought to check whether Canonical Livepatch service is working fine after upgrading to Ubuntu 18.10 from Ubuntu 18.04. I know it was meaningless since it was working fine in Ubuntu 18.04. But to my surprise when I ran   sudo canonical-livepatch status --verbose   I got the error:   2018/11/03 23:11:05 error executing status: Livepatchd error: The platform Ubuntu 18.10 is not supported. exiting.   I am now confused, is the problem only on my system? OR Is Canonical Livepatch service not supported on Ubuntu 18.10?  I would therefore like to request my fellow Ubuntu 18.10 users to do the check and if someone knows the reason/solution please reply to this thread. I dont need this vertices and/or edge. What is the best way to get rid of them? Use Data From Class Without A New Init Why do Star Trek vessels always approach each other on parallel planes? Space is just open 3D space, right? In Star Trek, ships are always coming from somewhere else: different solar systems, galaxies, etc. However, when they meet, they always seem to be on parallel planes (facing each other).  In some episodes I recall seeing ships that were adrift (and they appeared facing a different trajectory). How can there possibly be some standardized reference plane in space? Do ships just choose to align with each other to be polite? Or perhaps when entering a solar system, the pilot, or navigation system aligns orthogonally with the local orbital axis? Event binding on dynamically created elements? $ df -kh Filesystem      Size  Used Avail Use% Mounted on udev            7,8G     0  7,8G   0% /dev tmpfs           1,6G   19M  1,6G   2% /run /dev/sda2       622G  227G  364G  39% / tmpfs           7,8G  1,1G  6,7G  14% /dev/shm tmpfs           5,0M  4,0K  5,0M   1% /run/lock tmpfs           7,8G     0  7,8G   0% /sys/fs/cgroup /dev/loop1      143M  143M     0 100% /snap/inkscape/2527 /dev/loop3       79M   79M     0 100% /snap/core/1441 /dev/loop0       79M   79M     0 100% /snap/core/1577 /dev/loop4       79M   79M     0 100% /snap/core/1512 /dev/loop2       62M   62M     0 100% /snap/pencilsheep/5 tmpfs           1,6G   92K  1,6G   1% /run/user/1000   What are these /dev/loop device files? My RAM is always full. Is this related? LINQ on a LinkedList - iterate over LinkedListNode, not T Why don't we read ''k'' here?  Know → /nəʊ/   Knotty → /ˈnɒt.i/   Knockout → /ˈnɒk.aʊt/ Prove that the projection $p_i$ from the product space $X_I S_i \tau$ into $S_i \tau_i$ is open for each $i \in I$ Full question: A function $f$ from a space $X, \tau$ to a space $Y, \tau'$ is said to be open if $f(V)$ is open in $Y$ whenever $V$ is an open subset of $X$. Prove that the projection $p_i$ from the product space $X_I S_i \tau$ into $S_i \tau_i$ is open for each $i \in I$.  Attempt: Let $S = \Pi_{i \in I} S_i$ be the product space. Then, let $p_i: S, \tau \to S_i, \tau_i$ be the projection $p_i(s) = s_i$ for each $s \in S$. Then, $p_i$ is continuous for all $i \in I$.  Then, if $V_i$ is open in $S_i$, then $p_i^{-1}(V_i) = S$ is one of the open sets in the basis, namely $\{\Pi_I V_i : V_i \text{ open in } S_i \text{ and } V_i = S_i \text{ for all but at most finitely many i}\}$, and we see clearly that $V_i$ fits the criteria for all $i \in I$ because we defined $V_i$ as being open in $S_i$ and $I$ is a finitely countable set, hence there will be at most finitely many $i$ for which $V_i = S_i$.  Thus, the projection $p_i$ is open for each $i \in I$.  I'm not sure if I tackled this correctly and wanted to make sure I was on the right track or got it right. I have a laptop that has Windows 8 which I use for playing Minecraft. However when using the inventory, I have to hold my mouse to drag and pick the items and I don't like that.   I have tried switching mouses but that didn't work, is this possible to fix? How can I list the contents of a .zip folder in c#? How can I list the contents of a zipped folder in C#? For example how to know how many items are contained within a zipped folder, and what is their name? Are there any tools, methods, incantations to recover recently deleted files on Ubuntu?  If it makes any difference, I want to recover a  2.x database file. But would be better to have a method/tool that works on any kind of file. Conditional probability and venn diagram How to select lenses for outdoor sports photography? I would like to get into sports photography, and would probably want a better telephoto lens so I can get better close up shots.  What other factors should I keep in mind as I make my choice? Does IS (or VR) matter?  What is a good focal length?  I'll probably be shooting mostly little league baseball/softball, church softball leagues, and kids soccer. What is the purpose of a serial number?