This is related to this question:  Let $A$ be a complex square matrix. In one of the answers, they used eigenvalues of $A^2$ to calculate eigenvalues of $A$. I have a question in terms of this. We can easily show that if $\lambda$ is an eigenvalue of $A$, then $\lambda^n$ is an eigenvalue of $A^n$. So, in set notation, If $\{\lambda_1,\cdots \lambda_k\}$ is a set of eigenvalues of $A$, $\{\lambda_1^n,\cdots,\lambda^n_k\}\subset \{\textrm{Set of eigenvalues of }A^n\}$ holds. My question is, does $\{\lambda_1^n,\cdots,\lambda^n_k\}=\{\textrm{Set of eigenvalues of }A^n\}$ hold? If so, how can we prove it?
Theorem 13.9 in these  is the polynomial spectral mapping theorem:     Theorem 13.9 For a polynomial $p$ we have $\sigma(p(T)) = p(\sigma(T))$.     My questions:   First direction: Why would $q(T) (p(T) - p(\lambda) I)$ be an inverse for $T - \lambda I$? Update: @RobertIsrael said that instead $(p(T) - p(\lambda) I)^{-1} q(T)$ should be $(p(T) - p(\lambda) I)^{-1}$ be? According to the lines in the proof above, we have $$ (p(T) - p(\lambda) I)^{-1} = ((T - \lambda I) q(T))^{-1} \overset{?!}{=} q(T)^{-1} (T - \lambda I)^{-1}, $$ but as $\lambda \in \sigma(T)$ by assumption, $(T - \lambda I)^{-1}$ doesn't exist, right? For the second direction: Why does $x_i \in \sigma(T)$ imply that $p(x_i) - \mu = 0$? I thought that $x_i \in \sigma(T)$ only implies that $T - x_i I$ is not boundedly invertible. Since $\mu \in \sigma(p(T))$ we have that $p(T) - \mu I = c \prod_{k = 1}^{n} (T - x_i I)$ is not boundedly invertible but how does that help?   In both statements there seem to be a notion of "a product of operators is not invertible iff all factors aren't invertible", but how can that be true? Consider i.e. shift operators on a sequence spaces such that $ST = I$, i.e. the left and right shift operators, then the above mentioned notion doesn't hold, right?
This problem is quite easily solved by using logarithms and derivative and forming the function $f(x,y) = x^y - y^x$, however there are assertions that this problem can be solved without using either of the two. I can not see how one would proceed to solve this problem in the absence of logarithmic manipulation and differentiation. Can anyone shed some light on this? I know that its is the possessive and it's is the contraction, and know when to use them.  But why doesn't the possessive have an apostrophe?   "The bear's eating a fish." [contraction] "The bear's coat is brown." [possessive] "It's eating a fish." [contraction] "Its coat is brown." [possessive] "One's eating a fish." [contraction] "One's coat is brown." [possessive]    lists the etymology as "From  +‎ ", and  says that this is actually the original form:     Originally written it's, and still deliberately spelled thus by some writers until early 1800s.   So what happened to the apostrophe?  When did people stop using it, and why did they?  It seems that it's as the possessive is more natural, as most people do this until they're taught that it is wrong (or even after).  Update: Online Etymology Dictionary has been updated to include two potential explanations:     The apostrophe came to be omitted, perhaps because it's already was established as a contraction of it is, or by general habit of omitting apostrophes in personal pronouns (hers, yours, theirs, etc.).   Can anyone back up either of these arguments?  The possessive one's still has the apostrophe, despite these. Using GParted to remove windows partition This occurs in the  released recently to the public. When someone posts a code block in their question or answer, the code block is cut off on the right hand side, and I can't scroll to the right to view the rest of it. How can I view these code blocks? Why does println(array) have strange output? ("[Ljava.lang.String;@3e25a5") Some prestigious universities like Caltech, Stanford, MIT and Berkeley never disclose their information regarding their cut off major and cumulative GPA for M.Sc and PhD programs in engineering. I also aware of the fact that graduate schools will look at the whole package and make their decision based on sets of factors like letter of recommendation, statement of purpose as well as GPA.   However, since application fee is pretty high it is not worth it to apply to schools that you know you have a slight chance of getting into it so, I would like to know what would be the acceptable GPA (Both major and Cumulative) for applicants to be competitive for top five schools in the world? Imagine that you have the following list. keys = ['name', 'age', 'food'] values = ['Monty', 42, 'spam']  What is the simplest way to produce the following dictionary? a_dict = {'name': 'Monty', 'age': 42, 'food': 'spam'} The parametric equation  $$\begin{align*} x(t) &amp;= \cos t\\ y(t) &amp;= \sin t \end{align*}$$  traces the unit circle centered at the origin ($x^2+y^2=1$).   Similarly,   $$\begin{align*} x(t) &amp;= \cosh t\\ y(t) &amp;= \sinh t \end{align*}$$  draws the right part of a regular hyperbola ($x^2-y^2=1$).  The hyperbolic trigonometric functions are very similar to the standard trigonometric function.  Do similar functions exist that trace parabolas (because it is another conic section) when set up as parametric equations like the above functions?  If so, are they also similar to the standard and hyperbolic trigonometric functions? Undergraduate research field different from PhD field I've got bunch of files with ill-formed numbering:  prefix-#.ext       | for files with number 1-9 prefix-##.ext      | for files with number 10-99 prefix-###.ext     | for files with number 100-999   Due to further processing I need all of their names to be in format: prefix-###.ext. Is there any easy way to do that? If I inherit from a base class and want to pass something from the constructor of the inherited class to the constructor of the base class, how do I do that?  For example, if I inherit from the Exception class I want to do something like this:  class MyExceptionClass : Exception {      public MyExceptionClass(string message, string extraInfo)      {          //This is where it's all falling apart          base(message);      } }   Basically what I want is to be able to pass the string message to the base Exception class. Every Group is a Fundamental Group Why would I buy a bond with a negative yield? The only reason that I can think of is because I expect the bond to appreciate in price thus offsetting the loss in yield. I could still end up with less money than I started with, if I hold the bond until maturity. Ultimately, the only scenario where I would want to buy a bond with a negative yield is because I anticipate deflation and even lower rates in the future. Is this correct, or is there another reason?  Currently, a lot of the global government bond market is trading on negative yield, and there are even some corporate bonds that are too. How do I replace the *first instance* of a string in .NET? Why this JSFiddle does not work If, one fine day, someone found a contradiction in ZFC (or even ZF), what implications would such an event have for mathematicians? Is there currently any backup axiomatic system on par with ZFC that would enable mathematicians to continue pursuing mathematics, should the unthinkable occur? How can I copy/transfer games via external hard drive? I have successfully transferred "Killing Floor" from my brother's laptop to my own via external hard drive. The problem, now that I have all the files, is that I don't know how to launch the game through Steam. All I did was copy files from his Laptop to the hard drive, then from the hard drive to my laptop, then I put the files in the "Steamapps" folder under "Steam" in my laptops hard drive. So the question is, how do I start up the game through Steam? How do you express that your understanding of something is completely changed? When something blows your mind and changes your understanding of something that you've believed in for years, how do you express it (formally or informally)? I will be writing scripts for ArcGIS Desktop in Python with ArcPy.   Is there a community or open source project where code and models can be shared? Prevent malicious bots from posting spam I remember  due to misuse and I wonder if bots have a part of it. If the bot is POSTing something to my site what are ways I can combat it? I was thinking of setting some cookies and having the cookies changed via JavaScript + timestamp and sign (so yesterday's cookies can't be used today and next week).  I'm sure most people/bots would just use another site instead of enabling JavaScript in their bot.  What else can I do? I'm thinking daily POST limit and a honeypot for generic bots who just randomly post spam.