Find the minimal polynomial of the $n$-dimensional matrix $(a_{ij})$ when the matrix elements $a_{ij}$ have the form $a_{ij} = u_i v_j.$    Let $A=uv^T$ where $u,v$ are column vectors.  Then rank$(A)\leq$rank$(u)\leq1.$ So kernal$(A)\geq n-1.$ That is, the geometric multiplicity $\geq n-1.$ According to Jordan decomposition theorem, the number of Jordan blocks w.r.t. $0\ \geq n-1.$ Therefore, the algebraic multiplicity of $0$  $\geq n-1.$  Suppose rank$(A)=1,$ how do I find the other eigenvalue?
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
Why does a bigger sensor lead to a shallower depth of field? Tractatus 3.333 and Russell's paradox Can anyone explain to a non-logician how Tractatus 3.333 refutes (or fails to refute) Russell's Paradox? Please explain his use of symbols! Superscript outside math mode Importing an .obj file and nothing is showing Killing form on $\mathfrak{sp}(2n)$ Words for "a bit sad" and "a bit happy"? My earliest PhD graduation date is 10 months away. Funding permits me to stay 1–3 months longer, so the final date is still uncertain.  It is never too early to make contacts for possible post-docs. But how long in advance is too early to formally apply to post-doc job openings or funding opportunities? Depending on the bureaucracy, the path from applying to starting can probably be anything from a week up to a year — someone I know is applying with  which had the opening advertised in August, application deadline in October, will have interviews in December/January, and the commencement of work only in September, 13 months after the initial job advertisement. My personal guess is that the long time between interviews and starting is to create a level playing field between those who do and those who don't need to go through a visa application procedure.  If I'm too early in applying for a post-doc, a hosting institution might prefer someone who can start earlier and where the graduation date is more certain. For example, in my case the graduation date is still uncertain, so there is a risk in accepting even if they are willing to wait. Project funding might also require the money to be spent rather soon.  If I wait too long, there might be a considerable period where I'm between jobs. Having a long (>6 months) period between jobs is a disadvantage for me. Maybe it's also a disadvantage for a hosting institution (I'm not sure).  How do I find the balance? For example, considering the duration of funding and visa applications, does it make sense to apply e.g. ? Or is it in any case virtually inevitable to have at least several months before my PhD graduation and the start of a post-doc? Evaluate the limit of $\lim_{n\to\infty}\frac{1}{n^2}\left(\frac{2}{1}+\frac{9}{2}+\frac{64}{9}+\cdots+\frac{(n+1)^{n}}{n^{n-1}}\right)$ I am using a plugin to help "build" a customized product. There are different tabs/steps the user has to go through with back and continue buttons. At the last step I would like to add an "agree to terms" box with a checkbox "Agree". I have this done. But I would like to hide the add to cart button in Woocommerce until the user selects the checkbox agreeing to the terms.  This will be beneficial to me to make sure they reviewed the return policy as well as avoid confusion with the add to cart button appearing during all the steps, assuring they go through all the steps.  Here is the checkbox coding i pulled from console:  &lt;ul data-rules="{&amp;quot;Agree_0&amp;quot;:[&amp;quot;&amp;quot;]}" data-original-rules="{&amp;quot;Agree_0&amp;quot;:[&amp;quot;&amp;quot;]}" data-rulestype="{&amp;quot;Agree_0&amp;quot;:[&amp;quot;&amp;quot;]}" data-tm-validation="{&amp;quot;required&amp;quot;:true}" class="tmcp-ul-wrap tmcp-elements tm-extra-product-options-checkbox tm-element-ul-checkbox element_16 termsbox-ul"&gt;     &lt;li class="tmcp-field-wrap tmhexcolorimage-li-nowh"&gt;         &lt;label for="tmcp_choice_16_0_35"&gt;    &lt;input class="tmcp-field termsbox tmhexcolor_16_0_35 tm-epo-field tmcp-checkbox" name="tmcp_checkbox_16_0" data-limit="" data-exactlimit="" data-minimumlimit="" data-image="" data-imagec="" data-imagep="" data-imagel="" data-image-variations="[]" data-price="" data-rules="[&amp;quot;&amp;quot;]" data-original-rules="[&amp;quot;&amp;quot;]" data-rulestype="[&amp;quot;&amp;quot;]" value="Agree_0" id="tmcp_choice_16_0_35" tabindex="35" type="checkbox"&gt;     &lt;span for="tmcp_choice_16_0_35"&gt;&lt;/span&gt;&lt;span class="tc-label tm-label"&gt;Agree&lt;/span&gt;&lt;/label&gt; &lt;span class="price tc-price  hidden"&gt;     &lt;span class="amount"&gt;$0.00&lt;/span&gt; &lt;/span&gt;&lt;i data-tm-tooltip-html="This is were the description will be to agree" class="tm-tooltip tc-tooltip tcfa tcfa-question-circle"&gt;&lt;/i&gt;     &lt;/li&gt;&lt;/ul&gt;   Here is the add to cart section with the quantity  &lt;div class="quantity buttons_added"&gt;     &lt;input value="-" class="minus button is-form" type="button"&gt;    &lt;input class="input-text qty text" step="1" min="1" max="9999" name="quantity" value="1" title="Qty" size="4" pattern="[0-9]*" inputmode="numeric" type="number"&gt;     &lt;input value="+" class="plus button is-form" type="button"&gt;  &lt;/div&gt; &lt;button type="submit" name="add-to-cart" value="267" class="single_add_to_cart_button button alt"&gt;Add to cart&lt;/button&gt;   I am self taught with html and css, but javascript is to foreign for me  I tried the following in CSS but it does nothing.Tried is on codepen and works (i know needs to show instead of hide when checked)....  input[type=checkbox]:checked + .single_add_to_cart_button {     display:none !important;  } Find an equation in spherical coordinates for the surface represented by the rectangular equation The rectangular equation is $$x^2+y^2-8z^2=0$$ $$x^2+y^2=8z^2$$ Know in the relationship between rectangular and spherical coords. we can manipulate our given to fit the form: $$x^2+y^2+z^2=9z^2$$ $$\rho=x^2+y^2+z^2, \space z=\rho\cos(\phi)$$ $$\rho^2=9\rho^2\cos^2(\phi)$$ $$1=9\cos(\phi)$$ $$\frac{1}{3}=\cos(\phi)$$ $$\arccos(\frac{1}{3})=1.23 \space rads$$ And so the equation in spherical coords. is $\phi=1.23$ I know my math is correct but I have the wrong answer so I'm not sure where I went wrong. 'Country have' vs 'country has' 1) Australia has won the World Cup Cricket.      2) Australia have won the World Cup Cricket.   Which sentence is correct? Also, I would like to know more contexts where 'country has' and 'country have' are used. Classification of automorphisms of projective space Let $k$ be a field, n a positive integer.  Vakil's notes, 17.4.B: Show that all the automorphisms of the projective scheme $P_k^n$ correspond to $(n+1)\times(n+1)$ invertible matrices over k, modulo scalars.   His hint is to show that $f^\star \mathcal{O}(1) \cong \mathcal{O}(1).$ (f is the automorphism. I don't if $\mathcal{O}(1)$ is the conventional notation; if it's unclear, it's an invertible sheaf over $P_k^n$) I can show what he wants assuming this, but can someone help me find a clean way to show this? I am asked to determine if a series converges or not:  $$\displaystyle\sum\limits_{n=1}^{\infty} \frac{(2^n)n!}{(n^n)}$$  So I'm using the $n$th root test and came up with $\lim_{n \to {\infty}}\frac{2}{n}\times(\sqrt[n]{n!})$ I know that the limit of $\frac{2}{n}$ goes to $0$ when $n$ goes to infinity but what about the $(\sqrt[n]{n!})$? jQuery autocomplete tagging plug-in like StackOverflow's input tags? What solutions accomplish the same auto-completion that SO uses for entering tags?  There are plugins that can handle one word but I haven't seen any that handle multiple words. Will something accelerate forever if a constant force is applied to it on a frictionless surface? If so how? I don't understand the fact that let's say hypothetically I was pushing a block on a frictionless surface, and I kept applying the same force on the object how it would keep accelerating! Wouldn't there have to be a certain velocity where it would stop getting faster? If not then what is actually acting on the object to cause it to accelerate infinitely? Are $_SESSION variables stored on the client or the server? How do I animate time? I need to externally limit a process/session to a certain number of cores.  Are there any other possibilities than CPU affinity (I don't like the need to specify the actual cores) and cgroups (hard to integrate into our project)? Java is the key here. I need to be able to delete files but users expect to be able to "undelete" from the recycle bin.  As far as I can tell this isn't possible.  Anyone know otherwise? What does: Error Broken count >0 mean?