Let $T_\mathbb{R}$ be a real $n \times n$ matrix, and let $T_\mathbb{C}$ be the same matrix but considered as a complex one. What is the easiest way to see that $\min_{T_\mathbb{C}} \in \mathbb{C}[t]$ equals the image of $\min_{T_\mathbb{R}} \in \mathbb{R}[t]$ under the tautological map $\mathbb{R}[t] \to \mathbb{C}[t]$?
Like the title says, can a matrix in $\mathbb{R}$ have a minimal polynomial that has coefficients in $\mathbb{C}$?  I have a feeling the answer is yes because the characteristic polynomial can have coefficients in $\mathbb{C}$ and so I don't see any reason why not, I was merely asking in case there was some sneaky proof that proves that this cannot happen.
I need another LP-E5 type battery for my Canon 450D. Should I buy an original Canon battery or one of the many generic brands available? There are loads of cheapos on , for example.  I should add that the new battery will be added to my existing Canon battery in a battery grip.  Does anyone have any experience (good or bad) of using generic batteries? Intertext vertical spacing How do I reduce the vertical spacing of \intertext? Tikz shortandoff-on newcommand problem Usually I have no problem with babel and !:, and I don't use shortandoff-on switch. But since I made a command using !#1 parameter, the problem comes up again.  The same synthax used outside the command work fine, but in the command I get the error message :  ! Package tikz Error: + or - expected.  How to solve the problem without using the shortandoff-on switch ?  \documentclass[french,a4paper]{article} \RequirePackage{etex} \RequirePackage[utf8]{inputenc} \RequirePackage[T1]{fontenc} \RequirePackage{fourier}  \RequirePackage{tikz} \usetikzlibrary{calc,intersections}  %%%%%%%%%%%%%%%%%%%%%% %%%% Francisation %%%% %%%%%%%%%%%%%%%%%%%%%%  \RequirePackage{babel}  %%%%%%%%%%%%%%%%%%% ---------------------------------------- %%%% Géométrie %%%% %%%%%%%%%%%%%%%%%%%-----------------------------------------      % #1 option du path     % #2 premier point     % #3 second point     % #4 longueur dans un sens     % #5 longueur dans l'autre     % défini Mil#1#2 \newcommand{\Med}[5][]{%     \path[#1,name path=Med#2#3] ($(#2)!.5!(#3)$) node (Mil#2#3) {}      ($(Mil#2#3)!#4!90:(#3)$) -- ($(Mil#2#3)!#5!270:(#3)$) ; }  \tikzset{small dot/.style={fill=black,circle,scale=0.3}}  \begin{document}  %\shorthandoff{:!} \begin{tikzpicture}  \coordinate[label=below left:$A$] (A) at (0,0) ; \coordinate[label=below right:$B$] (B) at (4,0) ;  \draw (A)--(B) ;  %\Med[draw]{A}{B}{1}{1} ;  \draw($(A)!.5!(B)$) node (MilAB) {}      ($(MilAB)!1!90:(B)$) -- ($(MilAB)!1!270:(B)$) ;   \end{tikzpicture} %\shorthandon{:!}  \end{document} The mystery of the tagged answer - and more It is (currently)   to tag answers. Yet, if I query , then  does have a tag, namely javascript. I'm at a loss: how did that happen?&dagger;  SELECT Id, Tag FROM Posts, PostsTags  WHERE Posts.PostTypeId = 2 AND Posts.Id = PostsTags.PostId   Edit: The are also five answers (, , ,  and ) with a view count of 1 instead of 0:  SELECT Id, ViewCount FROM Posts WHERE PostTypeId = 2 AND ViewCount &lt;&gt; 0   These results also hold through at least January 10.    &nbsp; &dagger;: PostTypeId = 2 means that the post is an answer, not a question (PostTypeId = 1). A positive polynomial is the sum of two squares in $\mathbb{R}[X]$ Let $P\in\mathbb{R}[X]$ be a positive polynomial.  I want to show that there exists $A,B\in\mathbb{R}[X]$ so that $P=A^2+B^2$    $\displaystyle P=a\prod_{k=0}^q(x-a_k)\prod_{k=0}^{2p}(x-b_k)=a\prod_{k=0}^q(x-a_k)\prod_{k=0}^p(x^2-2\Re(b_k)+|b_k|^2)$ where $a$ is the leading coefficient, $a_k$ are the real rooks, $b_k$ are the complex roots.  What can i do now ? I know that the identity $(a^2+b^2)(c^2+d^2)=(ac-bd)^2+(ad+bc)^2$ should come in handy, but I can't figure out :  1) How to show that the product real roots is a square  2) How to use the above identity for the complex roots.  If I can solve those two points, then I can solve the problem. Solving an equation over the reals: $ x^3 + 1 = 2\sqrt[3]{{2x - 1}}$ Solve the following equation over the reals:$$ x^3  + 1 = 2\sqrt[3]{{2x - 1}} $$ I noticed that 1 is a trivial solution, then I tried raising the equation to the 3rd, then dividing the polynomial by $(x-1)$.. But I can't see the solution, how do I go from here? Below is my code to create the component dynamically - The component is "ui:inputselect"  $A.createComponent(             RecoveryLineItem.Attributes[i].name,{                 "class" : "slds-input ",                 "aura:id" : 'ExampleId',                 "label" : RecoveryLineItem.Attributes[i].label,                 "multiple" : RecoveryLineItem.Attributes[i].multiple             }, ....   ..  console.log(component.find('ExampleId').get("v.value"))    Or console.log(component.find('ExampleId'))  is Giving as 'Undefined"  How to access the aura:id of the dynamically created components? Media keys not working for iTunes (pause/play, skip buttons) I had to write the proof to show that an Ideal $P$ of a commutative ring $R$ is prime Ideal if $R/P$ is an integral domain.       let $a,b\in R$ s.t. $ab\in P$ ,        $$ab+P=P\implies(a+P)(b+P)=P\implies\overline a \overline b=0$$  and as $R/P$ is an integral domain either $a=0$ or $b=0$ $\implies a+P=P$ or  $b+P=P$.        $\therefore$ either $a\in P$ or $b\in P$.   Hence, $P$ is a prime Ideal.    Am I correct in writing this or am I making some error... How to create a Java application which can be run by a click? Lightning Web Components: navigate to the 'new' object page passing in specific record type Id? I am using LWCs to display lightning cards in a list.  The list shows information about Programs.  Each Program card has a button, and this button needs to take the user to the 'new object' screen of the Application object.  Here is my code:   applyToProgram(event) {     // The recordTypeId is coming from the .html     // I have verified that the value is correct.     var rtId = event.target.id;      // How do I use rtId below?     this[NavigationMixin.Navigate]({         type: 'standard__objectPage',         attributes: {             objectApiName: 'Application__c',             actionName: 'new',             nooverride: '1',             recordTypeId: rtId  // This does not work         }     }); }   This is not working.  The user is presented with the 'new object' screen for the object's default record type and NOT the one specified in rtId.  I tried putting 'nooverride' and 'recordTypeId' in the state:{} argument, and I also renamed 'recordTypeId' to just 'recordType'.  None of these worked.  Can anyone help me with the correct syntax?  Is this even possible in LWCs?  Thanks, Jeff How to shrink Windows 7 boot partition with unmovable files I have just bought an HP laptop with Windows&nbsp;7 (64 bit). It has a 500&nbsp;GB HDD with three partitions: a small hidden system partition, a 12&nbsp;GiB HP recovery partition, and a 450&nbsp;GiB C: boot partition.  I would like to split this large C: partition into two partitions, leaving only 100&nbsp;GiB for the system, and giving the rest to a new data partition.  Although the Windows built-in Disk Management utility has an option to shrink the bootable partition, it only allows me to shrink it roughly by half, even though only 20&nbsp;GiB on the partition is used. As far as I understand, system unmovable files lie in the middle of the partition, preventing Disk Management utility to do what I want.    And since new HP laptops don't come with OS installation disks (they only allow you to create recovery disks yourself), I can't just repartition HDD and then reinstall OS.  So, is there a way to shrink the C: bootable partition and preserve Windows&nbsp;7 working?  P.S.: I have tried to use the third-party  utility, and after shrinking the partition, Windows&nbsp;7 stopped booting, with a . System recovery didn't work, and I had to do factory recover. Since this is a long process, I would like to avoid doing it again :) So, please, suggest only proven solutions. I'm about to use this diagram to make computing 3x3 determinants clear and easy for my students. I tried to add this picture to my lecture but it's blurred due to the bad quality of the picture. So i want to create exactly the same figure using latex.  I've searched and read a few topics about highlighting elements and diagonals in matrix but it does not work in this case. I want to add plus signs and minus signs as in the picture, and highlight elements in the matrice with elements outside the matrice.  I appreciate all your help and your advice. Thank you in advance ! How to get over "device or resource busy"? I tried to rm -rf a folder, and got "device or resource busy".  In Windows, I would have used LockHunter to resolve this. What's the linux equivalent? (Please give as answer a simple "unlock this" method, and not complete articles like . Although they're useful, I'm currently interested in just ASimpleMethodThatWorks™) I'm trying CTest in CMake in order to automatically run some of my tests using make test target. The problem is CMake does not "understand" that the test I'm willing to run has to be built since it is part of the project.  So I'm looking for a way to explicitly specify this dependency. How to cite own unpublished work in text and reference? Does metric signature affect the stress energy tensor? If one were to derive the stress-energy tensor for a metric with $(+,-,-,-)$ signature would it be different from the stress-energy tensor derived from the same metric but with $(-,+,+,+)$ signature? is this set a regular surface? I'm reading "Differential Geometry of Curves and Surfaces of Manfredo Docarmo" I'm doing  the exercises of the chapter 2. Here is the definition of regular surface that we are following:   I have problems with this exercise:   Is with the first part, I think that it's not true that it's a regular surface (the second it is). I don't know how to prove that something is not a regular surface. Remove all files/directories except for one file I have a directory containing a large number of files. I want to delete all files except for file.txt . How do I do this?  There are too many files to remove the unwanted ones individually and their names are too diverse to use * to remove them all except this one file.  Someone suggested using   rm !(file.txt)   But it doesn't work. It returns:  Badly placed ()'s    My OS is Scientific Linux 6.  Any ideas? If photon energies are continuous and atomic energy levels are discrete, how can atoms absorb photons? If photon energies are continuous and atomic energy levels are discrete, how can atoms absorb photons? The probability of a photon having just the right amount of energy for an atomic transition is $0$.