Matrix determinant bounding lemma I have two positive semidefinite matrices, $A$ and $B$, of size $((N-1) \times (N-1))$.  I know that for all $i$ and $j$ in $\{1,...N-1\}$, $|A_{i,j}-B_{i,j}|&lt;k$ for some positive real constant $k$ and $|A_{i,j}|&lt;l$ and $|B_{i,j}|&lt;l$ for some positive real constant $l$. How can I place on the below expression an upper bound in terms of $k$, $l$ and $N$?  $$|\det(A)-\det(B)|$$  I.e. I'm looking for an expression in terms of $N$ and $k$, $C_{n,k,l}$, such that:  $$|\det(A)-\det(B)|\leq C_{n,k,l}$$  I believe such an expression exists as I am working through a proof which does something similar but gives no explanation as to how it achieved it. Going by what is in the proof, it seems as though the answer should come to something similar to:  $C_{n,k,l}=k2^{N}N(N!)(l+1)^{N-1}/2$  The proof I am working through says this result should be obvious but I am unsure why.
Bound on the difference of two determinants Let $A$ and $B$ be two real, $n\times n$ matrices. Using Hadamard's inequality, it is not hard to show that $$ \left|\det A - \det B \right| \leq \|A-B\|_{2} \frac{\|A\|_{2}^n -\|B\|_{2}^n}{\|A\|_2 -\|B\|_2}. $$ Where $\|A\|_2=\sqrt{\sum_{i,j}a_{ij}^2}$.  From this, I can derive a sup bound, for example  $$ \left|\det A - \det B \right| \leq n^{n+1} \|A-B\|_{\infty} \max (\|A\|_{\infty}^{n-1},\|B\|_{\infty}^{n-1}). $$ Where $\|A\|_\infty=\sup_{i,j}|a_{ij}|$.   The constant $n^{n+1}$ is not the best bound possible : any reference (or proof) for a better (or the best) one? I show below that one can obtain $n^2(n-1)^{n-1}$, but that isn't much better. I just tried  $10^5$ random matrices on Maple and obtained a maximal constant (much) smaller  than one : this is not a proof, but it looks like there is room for improvement nevertheless.    Just for completeness (and in case someone sees a factor I missed), to get the first bound, writing $A=[A_1,\ldots,A_n]$ in terms of its column vectors, an expansion shows \begin{eqnarray*} \det A &amp;=&amp; \det (A_1 -B_1,A_2,\ldots,A_n) + \det (B_1,A_2,\ldots,A_n) \\   &amp;=&amp; \sum_{j=1}^n \det (B_1,\ldots, B_{j-1}, A_j -B_j,A_{j+1},\ldots,A_n) \\ &amp;&amp; + \det B,  \end{eqnarray*} Thus by Hadamard's inequality, \begin{eqnarray*} \det A -\det B &amp;\leq&amp;  \sum_{j=1}^n \prod_{i=1}^{j-1} \|B_i\|_{2}\prod_{i=j+1}^{n} \|A_i\|_{2} \|A_j-B_j\|_{2} \\ &amp;\leq&amp;  \|A-B\|_{2} \sum_{j=1}^n \|B\|^{j-1}_{2}\|A\|^{n-j}_{2} \\  &amp;=&amp; \|A-B\|_{2} \frac{\|A\|_{2}^n -\|B\|_{2}^n}{\|A\|_2 -\|B\|_2}. \end{eqnarray*}  The second bound is just that $x^n -y^n\leq n \max(|x|^{n-1},|y|^{n-1}) |x-y|$ and $\|A\|_2 \leq n\|A\|_\infty$.    Another approach is calculus, namely, to write that $\det B - \det A = f(1)-f(0)$, with $f(t)=\det(A + t(B-A))$.   By the mean value theorem $f(1)-f(0)\leq \max |f^\prime (t)|$.   We can compute that $$f^\prime(t) =  {\rm trace}\left({\rm Cofm}(A +t(B-A))(B-A)\right)$$  (if I did not mess up, using the formula for the differential of a determinant, where Cofm means the matrix of Cofactors).   Then, it should deliver something better, if there is a nice way to bound it. The simplest thing is to use Cauchy-Schwarz, namely $$ \left|{\rm trace}\left({\rm Cofm}(A +t(B-A))(B-A)\right)\right|\leq \|B-A\|_{2} \|{\rm Cofm}(A +t(B-A))\|_{2} $$ and then, by lack of a better idea,  $$ \|{\rm Cofm}(A +t(B-A))\|_{2}\leq n \max_{ij} |{\rm Cof}_{i,j}(A +t(B-A))|, $$ and brutally, $|{\rm Cof}_{i,j}(A +t(B-A))|\leq ((n-1) \max(\|A\|_\infty,\|B\|_\infty))^{n-1}$ gives a slightly better constant, namely $$ n^2(n-1)^{n-1} &lt;n^{n+1} $$ but that still seems a very rough way to bound a determinant, as it is never sharp, since to attain this bound all coefficients should be equal, and therefore the cofactor would be zero. They are of the same order in $n$, and I suspect this order is wrong.
Can a multi-class Bard/Wizard write spells known from Bard, into Wizard spellbook? The expected value of $1/(1+X)$ where $X$ has geometric distribution Let $X$ be a random variable with a geometric distribution with parameter $p$.       Calculate $\Bbb E[1/(1+X)]$   I've looked at answers online but still do not understand how to solve this.   Specifically, I do not understand how to to get from $$\frac{p}{1-p} \sum_{x=0}^\infty (1/(1+x))((1-p)^x+1)$$  to  $-p\ln(p)  /(1-p)$. Search window not opening in ArcMap? How do I check if an integer is even or odd? I tried to write out the aura if statement but I am getting stuck if someone can explain it to me in detail. thank you  &lt;aura:if isTrue=&quot;{!lessthan(v.Tl.time, v.Tl.End_Date_Time1__c)}&quot;&gt;                            &lt;/aura:if&gt; Einstein equation  where $E$ is energy, $m$ mass, and $c$ the speed of light in vacuum. So does it mean objects without any mass does not posses energy for eg lights photons does not have mass but how can they posses energy How many Java string objects created in the code String s="abc"+"xyz";? Knowing $|z|$ and $|w|$ (and possibly $z\overline{w}$), what are $|z+w|^2$, $|zw|^2$, $|z-w|^2$, $|z/w|^2$? Question 1: Let $z$ and $w$ be complex numbers satisfying $|z| = 4$ and $|w| = 2$. Then enter in the numbers $$ |z+w|^2, |zw|^2, |z-w|^2, \left| \dfrac{z}{w} \right|^2 $$ below, in the order listed above. If any of these cannot be uniquely determined from the information given, enter in a question mark.  I plugged in $4$ and $2$ to each equation and got $36$, $64$, $4$, $4$, for question one. It was wrong.  Question 2: Let $z$ and $w$ be complex numbers satisfying $|z| = 5, |w| = 2,$ and $z\overline{w} = 6+8i.$ Then enter in the numbers $$ |z+w|^2, |zw|^2, |z-w|^2, \left| \dfrac{z}{w} \right|^2 $$ below, in the order listed above. If any of these cannot be uniquely determined from the information given, enter in a question mark.  I don't know how to even start on question 2 because of the $z\overline{w}$. What happens to the temperature when an ideal gas is compressed? I am rather confused about this. I know from Charles' law that under constant pressure, the volume of a gas is directly proportional to its absolute temperature i.e.  $$\frac{V_1}{T_1} = \frac{V_2}{T_2}$$  Thus accordingly, during compression, temperature of the gas would decrease. But in Lectures of physics ,vol 1 by Feynman, it is written:     Suppose that the piston moves inward, so that the atoms are slowly compressed into a smaller space. What happens when an atom hits the moving piston? Evidently it picks up speed from the collision. [...] So the atoms are "hotter" when they come away from the piston than they were before they struck it. Therefore all the atoms which are in the vessel will have picked up speed. This means that when we compress a gas slowly, the temperature of the gas increases.   (Constant pressure?) So, this is contrary to Charles' law. Why does this happen? Who is right? Or are they both correct? I am confused. Help. Are there any massless (zero invariant mass) particles carrying electric charge?  If not, why not? Do we expect to see any or are they a theoretical impossibility? Show that $\ker \hat{T} = \text{ann}(\text{range } T)$ Do "owed" and "due" mean the opposite? How to schedule a task to run when shutting down windows Is a company e.g. Pfizer, singular or plural? Here is from an article from the  magazine:     One drug he talks about is an antidepressant called reboxetine (branded as Edronax by   its makers, Pfizer) that he used to prescribe to his patients.   Pfizer is a drug company. It is a singular noun. Why is it described as its makers, which is plural? I'd like to work on a bbcode filter for a php website. (I'm using cakephp, it would be a bbcode helper) I have some requirement.    Bbcodes can be nested. So something like that is valid.  [block]       [block]       [/block]       [block]           [block]           [/block]       [/block]   [/block]     Bbcodes can have 0 or more parameters.     Exemple:     [video: url="url", width="500", height="500"]Title[/video]   Bbcodes might have mutliple behaviours.     Let say, [url]text[/url] would be transformed to [url:url="text"]text[/url] or the video bbcode would be able to choose between youtube, dailymotion....  I think it cover most of my needs. I alreay done something with regex. But my biggest problem was to match parameters. In fact, I got nested bbcode to work and bbcode with 0 parameters. But when I added a regex match for parameters it didn't match nested bbcode correctly.   "\[($tag)(=.*)\"\](.*)\[\/\1\]"  // It wasn't .* but the non-gready matcher  I don't have the complete regex with me right now, But I had something that looked like that(above).  So is there a way to match bbcode efficiently with regex or something else.  The only thing I can think of is to use the visitor pattern and to split my text with each possible tags this way, I can have a bit more of control over my text parsing and I could probably validate my document so if the input text doesn't have valid bbcode. I could Notify the user with a error before saving anything.  I would use sablecc to create my text parser.   Any better idea? or anything that could lead to a efficient flexible bbcode parser?  Thank you and sorry for my bad english... How does the drift velocity of electrons in a conductor depend on the temperature? How does the drift velocity of electrons in a conductor depend on the temperature?  I have two contradicting views for this.    First, we can say that increasing the temperature of the conductor will increase the kinetic energy of the electrons. Hence, their drift velocity should increase with increase in temperature.  Or, from the relation $v_d = \frac{eE}{m}T$ ($T$ is the relaxation time) we can say that the drift velocity is directly proportional to the relaxation time. Increasing the temperature will obviously decrease the relaxation time - as collisions will become more frequent - and thus decrease the drift velocity. Hence, an increase in the temperature will cause a decrease in the drift velocity.   So which view is correct? How can a distribution have infinite mean and variance? It would be appreciated if the following examples could be given:   A distribution with infinite mean and infinite variance. A distribution with infinite mean and finite variance. A distribution with finite mean and infinite variance.  A distribution with finite mean and finite variance.   It comes from me seeing these unfamiliar terms (infinite mean, infinite variance) used in an article I am reading, googling and reading a thread on the , and not finding it a sufficiently clear explanation. I also haven't found any explanations in any of my own textbooks. How do I change my Minecraft username during the 30-day restriction? "errno 5 - input/output error" when trying to install Ubuntu Today I downloaded Ubuntu for my laptop. It runs great from a bootable USB, but when I tried to install it, I got the following error message: The installer encountered an error copying files to the hard disk:  [Errno 5] Input/output error  This is often due to a faulty CD/DVD disk or drive, or a faulty hard    disk. It may help to clean the CD/DVD, to burn the CD/DVD at a lower   speed, to clean the CD/DVD drive lens (cleaning kits are often    available from electronics suppliers), to check whether the hard disk   is old and in need of replacement, or to move the system to a cooler   environment.     I tried everything to install it on my laptop including redownloading the Ubuntu ISO image, but nothing worked. Command for taking a screenshot to a new file