Commutative matrices - approximation by commutative simultaneously diagonalizable matrices. It is well known that the set $\mathcal{D}_n(\mathbb{C})$ of all complex, diagonalizable, $n \times n$ matrices is dense in $\mathcal{M}_n(\mathbb{C})$, the set of all complex $n \times n$ matrices. And that two diagonalizable matrices which commute are simultaneously diagonalizable.   The exercise, which I am having difficulty to solve, is the following.   Let $A,B \in \mathcal{M}_n(\mathbb{C})$. Suppose $AB = BA$ and let $\varepsilon &gt;0$. Show that there exists two simultaneously diagonalizable complex, $n \times n $ matrices, $C$ and $D$ such that $\| A - C \| \leq \varepsilon $ and $\| B - D \| \leq \varepsilon$.   I have managed to find a solution when $A$ (or $B$) is diagonalizable... But I really cannot find the answer in the general case.   Anyone able to help me out ?
Approximating commuting matrices by commuting diagonalizable matrices Suppose the matrices $A$ and $B$ commute. Do there exists sequences $A_n$ and $B_n$ of matrices such that   $A_n \rightarrow A$, $B_n \rightarrow B$.  Each $A_n$ is diagonalizable  and the same for each $B_n$.  For every $n$, $A_n$ commutes with $B_n$.    Moreover, it would be nice if the following property was additionally satisfied: if $A,B$ are real, then $A_n,B_n$ can be chosen to be real as well.
I'm calculating the covariance of a distribution in parallel and I need to combine the distributed results into on singular Gaussian. How do I combine the two?  Linearly interpolating between the two almost works, if they are similarly distributed and sized.   provides a forumla at the bottom for combination but it doesn't seem right; two identically distributed distributions should have the same covariance, but the formula at the bottom of the page doubles the covariance.  Is there a way to combine two matrices? How to achieve nomenclature entries like: symbol, Description, Dimension and unit, etc? The title is pretty much I want to ask. More detail, the nomenclature entries should be sorted in groups like:   Nomenclature     Acronyms     Symbol  Description     % no Dimensions and Units      ...     Greek Symbols     Symbol  Description             Dimensions     Units     ...     Roman Symbols     Symbol  Description             Dimensions     Units     S       Surface area vector     L^2            m^2     ...    Superscripts     Symbol  Description     % no Dimensions and Units      ...    Dimensionless Numbers     Symbol  Description     Definition     ...   e.g. expected entries like this:       My current nomenclature settings for grouping [book class]:  %---------------------------------------------------------- %                        Nomenclature %---------------------------------------------------------- %\renewcommand{\nomname}{List of Symbols and Abbrev.} \renewcommand\nomgroup[1]{%   \ifthenelse{\equal{#1}{A}}{%    \item[\textbf{Acronyms}] }{%                  A - Acronyms     \ifthenelse{\equal{#1}{R}}{%      \item[\textbf{Roman Symbols}]}{%            R - Roman       \ifthenelse{\equal{#1}{G}}{%         \item[\textbf{•} Symbols }]}{%          G - Greek           \ifthenelse{\equal{#1}{S}}{%            \item[\textbf{Superscripts  }]}{{%          S - Superscripts         \ifthenelse{\equal{#1}{U}}{%          \item[\textbf{Subscripts }]}{{%                 U - Subscripts         \ifthenelse{\equal{#1}{X}}{%          \item[\textbf{Other Symbols }]}%            X - Other Symbols                     {{}}}}}}}}}} %\ifpdf \renewcommand{\nomlabel}[1]{\hspace{1 em}#1}%{\hspace{1.5 em}#1} \renewcommand*{\nompreamble}{\markboth{\nomname}{\nomname}}   But how to extend it with Dimensions and Units? How do I go about porting Ubuntu Touch to different devices? Why is ${x^{\frac{1}{2}}}$ the same as $\sqrt x $? Why is ${x^{\frac{1}{2}}}$ the same as $\sqrt x $?   I'm currently studying indices/exponents, and this is a law that I was told to accept without much proof or explanation, could someone explain the reasoning behind this.  Thank you. Let $G$ be a group and $H$ be a normal subgroup of index $p$ ( a prime ) ; suppose $K$ is a subgroup of $G$ not contained in $H$ , then is it true that $G=HK$ ? ( I know that the fact is true if $p=2$ , but I don't know for the general case . Please help . Thanks in advance ) How To quickly do derivatives with respect to matrices Whats a quick way to work the below (and problems similar) out?   For me to take this derivative it involves a lot of time and boring calculation, there has to be a better way.    This is taken from the solutions to Chris Bishop's Pattern Recognition and Machine learning: I have a simple question answer to which would help me more deeply understand the concept of (non)commutative structures. Let's take for example (our teacher's definition of) a ring:  Let $R\neq \emptyset$ be a set, let $\oplus:A\times A \to A$ and $\bullet :A\times A \to A$ be binary operations. Moreover, let $(R, \oplus)$ be a commutative group, $(R, \bullet)$ be a monoid and following property holds for all $a, b, c\in R$: $$a\bullet(b\oplus c) = (a\bullet b)\oplus(a \bullet c)$$ $$(b\oplus c)\bullet a = (b\bullet a)\oplus(c \bullet a)$$ Then ordered triple $\mathbf R = (R, \oplus, \bullet \mathbf)$ is called a (unitary) ring.  Moreover, we call ring $\mathbf R$ commutative iff $(R, \bullet)$ is a commutative monoid. Commutativity of a ring is always a matter of its multiplicative operation because the additive operation is always assumed to be commutative.  Could anyone explain me the bold part? Why do we even in non-commutative rings (and fields etc.) assume the addition to be always commutative? Is there some serious reason? Would it make any trouble? Or studying of structures with non-commutative addition just doesn't give us anything new so we can take addition as commutative simply because of our comfort? So my friend has a USB flash drive that isn't working. I don't know a whole lot about "recovery" software so I figured I'd ask you guys.  What steps can I try (I have access to a Linux machine as well) to see....if I can get ANYTHING off the flash drive?  I've google'd around, but most of the stuff seems for windows. For every vector space $V$ does there exist a linear functional $f$ ( a linear map from $V$ to $F$ the underlying field ) such that for some $ \vec v \in V$ , $f(\vec v) \ne 0$ ? If it does exist , can we prove the existence without the "axiom of choice " ? Is the existence equivalent to axiom of choice ? How to validate an email address using a regular expression? Over the years I have slowly developed a  that validates MOST email addresses correctly, assuming they don't use an IP address as the server part.  I use it in several PHP programs, and it works most of the time.  However, from time to time I get contacted by someone that is having trouble with a site that uses it, and I end up having to make some adjustment (most recently I realized that I wasn't allowing 4-character TLDs).  What is the best regular expression you have or have seen for validating emails?  I've seen several solutions that use functions that use several shorter expressions, but I'd rather have one long complex expression in a simple function instead of several short expression in a more complex function. the largest integer $n$ for which $n+5$ divivides $n^5+5$? When we divide $n^5+5$ by $n+5$, we get a remainder $-620$, i.e., $n^5+5=K(n+5)-620$ , now how to proceed further? I have a question pertaining to the concept of acceleration and it's formula - Both seem to give me different answers.  I was asked:     A train is moving at a velocity of $20\ \mathrm{m/s}$. It hits the breaks and starts slowing down at the rate of $8\ \mathrm{m/s^2}$. How long does it travel before coming to a stop?    $a=-8\ \mathrm{m/s^2}$ $u=20\ \mathrm{m/s}$ $v=0$ $s=?$ (Displacement is asked)   According to the formula of $s= \frac{(v^2-u^2)} {2a}$, answer I got was 25 meters.  BUT!  My concept of acceleration explains me this :    According to my intuition of acceleration and velocity, when  an object is said to be accelerating at $-8\ \mathrm{m/s^2}$, it means that with every passing second, it's velocity will reduce $8\ \mathrm{m/s}$. So,   At 1st second - Velocity = $20\ \mathrm{m/s}$ - It'll travel 20 meters. At 2nd second - $v = 20\ \mathrm{m/s}-8\ \mathrm{m/s}=12\ \mathrm{m/s}$ -  Train travels 12 meters At 3rd, it travels 4 meters. At 4th, it halts.   So total is 20+12+4 = 36 meters. It travels 36 meters before halt.  Where am I going wrong in my conceptual understanding, I really don't understand! Silencing "Your disk is almost full" notification Real definition of "countable set" Shapefile layer no longer editable (toggle edit option greyed out) in QGIS? 80s/90s book where boy stumbles on game where aliens gamble their planets I don't remember much about it other than the setting. It's about a boy who one night stumbles upon some different alien races playing a game in which their civilizations will end if they lose. He gets involved in the game somehow and is hunted. "1 in 10 are" or "1 in 10 is"? Take the examples:   "One in ten children are dyslexic."  "One in ten children is dyslexic." "One in ten children has dyslexia." "One in ten children have dyslexia."   The "one" is singular so 2 and 3 should be correct. But the "one in ten" is a fraction" so 1 and 4 should be correct. And yet I think I usually say 1 and 3. Which is/are correct?! Rectangle with cross on title bar I have installed Ubuntu 14.04 and I think that I have found a bug. When I open terminal and  go to edit profile preferences I get a rectangle with a cross on the title bar. It happened with other applications too. Here are the screen shots: Entire function problem: translation I am stuck in a question for Algebraic Structures:  I need to prove: If given any ring $R$, for any $a$ in $R$ $a\cdot a=0$ then $a+a=0$.  (of course $+$, $*$ and $0$ regarding $R$)  I didn't find any question regarding this question.