Symmetric positive semidefinite closedness Given $A,B$ symmetric positive semidefinite matrices of equal size it is clear $A^2,B^2,(A+B)^2,A+B$ are all symmetric positive semidefinite.  However is $2AB$ symmetric positive semidefinite?
Is the product of symmetric positive semidefinite matrices positive definite? I see on Wikipedia that the product of two commuting symmetric positive definite matrices is also positive definite. Does the same result hold for the product of two positive semidefinite matrices?  My proof of the positive definite case falls apart for the semidefinite case because of the possibility of division by zero...
How to know the usage of undocumented function like LinearAlgebra`BLAS? Placing point over pixel with maximum value falling in grid overlay using QGIS Complete view of where the PATH variable is set in bash $\{a_n\}$ be a sequence such that $ a_{n+1}^2-2a_na_{n+1}-a_n=0$, then $\sum_1^{\infty}\frac{a_n}{3^n}$ lies in... Let $\{a_n\}$ be a sequence of positive real numbers such that   $a_1 =1,\ \  a_{n+1}^2-2a_na_{n+1}-a_n=0, \ \ \forall n\geq 1$.   Then the sum of the series $\sum_1^{\infty}\frac{a_n}{3^n}$ lies in...  (A) $(1,2]$, (B) $(2,3]$, (C) $(3,4]$, (D)$(4,5]$.   Solution attempt:   Firstly, we figure out what $\frac{a_{n+1}}{a_n}$ is going to look like. We get, from the recursive formula, $\frac{a_{n+1}}{a_n}=1+\sqrt{1+\frac{1}{a_n^2}}$ (remembering the fact that $a_n&gt;0$, the other root is rejected).   We know that, if $\lim_{n \to \infty}\frac{a_{n+1}}{a_n}&gt;1$, then $\lim a_n \to \infty$.   Further, $(a_{n+1}-a_n)= \sqrt{a_n(a_n+1)}&gt;0$. (Again, the other root is rejected due to the same reason).   Hence, $(a_n)$ increases monotonically.   Therefore, the largest value of $\frac{a_{n+1}}{a_n}$ is approximately $1+\sqrt{1+\frac{1}{1}} \approx 2.15$  Now, the sum can be approximated as $\displaystyle\frac{\frac{1}{3}}{1-\frac{2.15}{3}} \approx 1.3$ (In actuality, $\mathbb{sum}&lt; 1.3$).  So, option $(A)$ is the correct answer.   Is the procedure correct?   I have been noticing a handful of this type of questions (based on approximations) lately, and the goal is to find out where the sum / the limit of the sequence might lie.   Is there any "definitive" approach that exploits the recursive formula and gives us the value, or does the approach varies from problem to problem? How can I write SQL for a table that shares the same name as a protected keyword in MySql? What is the best way to restore the state of ViewController I want to use a very long USB cable to place a webcam in a different room. The maximum length that I found was five meters.   What is the maximum length of a USB cable? What is the maximum length achievable when using USB extension cables? How to select distinct field values using Solr? How can I perform a t-test (or another hypothesis test) on NPS (Net Promoter Score) results? Here is an explanation of how NPS is calculated:    I'm interested in testing two net promoter scores to determine if they are statistically different. I read a great answer to calculating margin of error for NPS (see link below), but I'm really interested in testing to see if there is a difference between two scores, because I suspect that our results aren't as "different" from year to year as they appear to be.    Is this at all possible? I understand t-tests are typically used to test whether two different record sets are statistically different. But is it possible to test Net Promoter Scores, either with a t-test or some other hypothesis test?  Any ideas you have would be a great help. Thank you! I just found out about the Mesh Display : Length feature in the "N" panel (thanks to ) that allows one to measure the length of a face or vertice.  But my first encounter with this tool gives me complex feelings of love and hate, see how, as I was trying to measure the aspect ratio of this TV screen, I am getting WIDTH = 20.33 and HEIGHT = 37.57.  These values seem awfully wrong, since my betrusted eyes tell me the height should be shorter than the width.    I have tried measuring the single vertices as well, to identical results. Creating shapefile from current dataframe extents in layout view of ArcMap? Where is the ArcGIS 10 tool for creating a shapefile from current dataframe extents in layout view?  Have looked around and the closet thing i can find is Toolbox's Grid/Strip Map Index tools under Data Drive Pages.  I just want to be able to create a single polygon rectangle shp file based on the data frame (in layout view) for any given scale / page setup. Why should I use an Initialization Vector (IV) when I have unique keys? I took a look at  but my question is not the same.  I have unique keys encrypting each plaintext (in CBC mode, AES-256) and I do not use a key to encrypt more than one plaintext. Is it insecure to use null IVs in this case? If it is insecure, why? How to create two classes in C++ which use each other as data? Sign of the vector (especially $g$) while calculating I'm a high school student and I'm studying Newton's second Law. While my teacher is calculating a net force of an object he always treat all vectors as positive numbers. I think what he meant is the magnitude of the vector which is always positive, but the problem is that he didn't use the magnitude symbol, for example like $||Fg||$. He said the acceleration due to gravity , g, is positive, but shouldn't it be negative since it is pointing down. This also cause a problem: every time when I calculate the net force I can't add all forces together, instead I need to choose to use minus sign when the vector is pointing down. So in general which way is correct? How do you type a single backslash, code-formatted, in a comment? Is there a way to type something in a comment that renders like this?     You'll need to escape it with a backslash - that is, the \ character.   I've tried various combinations, but a single backslash formatted as code seems to be impossible. Can this be fixed, or can I be enlightened please?  And to clarify: I expect this:  You'll need to escape it with a backslash - that is, the `\\` character.   to work properly, but it doesn't. Instead, one backslash escapes the ` and I'm left with a mess. I need to install more applications - can I use my external HD (samsung T5) and add an additional applications folder? I have loads of space on it and it the speeds are great. How can I change pdf page size? I would like to split pages in pdf so that each page would be divided into 2 pages ( e.g. form 100 pages I would go into 200 but smaller - exactly half size).  I need this to read pdfs on my ebook reader.  Anybody knows how can this be done ? Inverse of a Positive Definite Let K be nonsingular symmetric matrix, prove that if K is a positive   definite so is $K^{-1}$ .   My attempt:  I have that $K = K^T$ so $x^TKx = x^TK^Tx = (xK)^Tx = (xIK)^Tx$ and then I don't know what to do next. What is the use for non-inverting input resistance in a negative feedback opamp? Here in the Figure R5=50K. This resistor value is not in the equations to obtain output voltage, and in many tutorials non-inverting input is directly grounded without a resistor. What might be the reason to use it here? How do I restrict my kids' computing time? Access to our computer (not only to the internet) needs to be restricted for the accounts of my kids (7, 8) until they are old enough to manage this by themselves. Until then we need to be able to define the following:   the hours of the day when computing is o.k. (e.g. 5 - 9 pm) the days of the week when computing is not o.k. (e.g. mondays to fridays) the amount of time allowed per day (e.g. 2 hours)   In 11.10 all of the following that used to do the job don't work any more:   : for > 11.10 no more available through the ppa. : command line alternative, but from 11.10 removed from the repositories. : Looks great but repeatedly crashes to force restarting X-server. So we can't use or recommed this program at the moment.   Are there any other alternatives?