Consider the $p \times p$ sample covariance matrix: $$\mathbf{S} = \frac{1}{n-1} \cdot \mathbf{Y}_\mathbf{c}^\text{T} \mathbf{Y}_\mathbf{c} \quad \quad \quad \mathbf{Y}_\mathbf{c} = \mathbf{C} \mathbf{Y},$$ where $\mathbf{C} = \mathbf{I}-\frac{1}{n} \mathbf{1} \mathbf{1}^\text{T}$ is the $n \times n$  and $\mathbf{Y}$ is an $n \times p$ matrix.  How can it be proved that if the variables are continuos, not linearly related and $n-1&gt; p$ then the sample covariance matrix is ​​positive definite? The following clue is found in Ranchera's book:
When computing the covariance matrix of a sample, is one then guaranteed to get a symmetric and positive-definite matrix?  Currently my problem has a sample of 4600 observation vectors and 24 dimensions.
How to remove ubuntu partitions and use them to install windows How can I bypass Activation Lock? I bought a used iPhone 4.  The previous owner re-set the phone, but did not remove the device from their iCloud.  It is asking me for their iCloud information when I try to set-up the phone.  I don't have contact with the previous owner anymore.  Is there a way to remove the device from their iCloud so I can use a new one? $2$ out of $3$ property of the unitary group this is a follow up to the problem described .  I have the same problem and the proposed solution does not completely solve the problem. I mean, it does solve the problem but also creates another one. The new problem is that the theorems, lemmas and the rest are placed in such as way that they are not cut (of course), but this leaves then big empty spaces in many pages. So the problem is that, the problem described  does not seem to be fully resolved using the nocut option.   Do you any of you have a good idea on how to solve this?  Thanks! I have a character playing a multiclass Barbarian/Monk. Does the resistance from rage combine with Deflect Missiles to negate the damage and make the reaction attack? Automatically detect edit abuse and exclude from badges/flag for mod review has 5 pages of 1 character edits (including an edit to one of my posts), apparently in a massive attempt to earn the Copy Editor badge.  Should Copy Editor, etc., ignore 1 character whitespace-only edits?    If we still want to count minor edits (which may be useful), could we at least detect edit abuse via some secret algorithm? ~20+ one character whitespace-only edits in the course of an hour should at least raise a flag of some sort.    If whitespace character are too hard to count, can we at least assume that 20 non-tag edits in 20 minutes (less than a minute per edit) is a good signal of abuse? The specific algorithm should be secret, obviously, but I think these sorts of edit abuses shouldn't be too difficult to detect. When I ssh into my Ubuntu 18.04 box I get the motd  1 update could not be installed automatically   Which goes on to tell me to check the unattended-upgrades log.  The log shows no problems. How can I clear this bogus message? How to get Ubuntu download for a Mac? I am looking for a way to get the output of a command when it is run from within a C++ program.  I have looked at using the system() function, but that will just execute a command. Here's an example of what I'm looking for:  std::string result = system("./some_command");   I need to run an arbitrary command and get its output. I've looked at , but I have not found anything that will give me what I need. If $\sigma_n=\frac{s_1+s_2+\cdots+s_n}{n}$ then $\operatorname{{lim sup}}\sigma_n \leq \operatorname{lim sup} s_n$ This is a question from the book Methods of Real Analysis by R. R. Goldberg.     If $(s_n)$ is a sequence of real numbers and if $$\sigma_n=\frac{s_1+s_2+\cdots+s_n}{n}$$ then prove that: $\operatorname{{lim sup}}\sigma_n \leq \operatorname{lim sup} s_n$.   I don't have any idea how to start working on this problem. Please help. Thanks. I am a newbie in linux and I don't know how to open the Ubuntu Software Center via Terminal!! Please help The earth is not a sphere, because it bulges at the equator.   I tried fiddling with centripetal force equations and gravity, but I couldn't derive why this bulge occurs.    Is there   (a) a mathematical explanation using forces (not energies) and   (b) a simple intuitive explanation to explain to others why the bulge occurs? Changing units in QGIS: degrees to meters I have a global map with CRS WGS84-EPSG:4326 and I want to calculate the area of my polygons. From what I read, to change the unit from degrees to meters I need to change my CRS. However, there is no global system for this. Is there any solution to change the unit from degrees to meters for a global map? If I push or hit an object in space will it rotate or move along a straight line? If I push or hit an object in space (vacuum and no gravitation) in direction what is not going trough its centroid, will it rotate or move along in straight line?   I expect that on earth it will depend on what is less difficult for the object (rotation or linear movement). So the object will do some kind of combination of both movements (rotating and also moving along the direction of impulse or force).  But how could an object "decide" what to do in space, where is not resistance? Plural nouns in nominal compounds I wonder whether there are rules or guidelines regarding plural nouns in nominal compounds. For example a compound university students list. If there are many lists and many universities is it grammatically correct to say universities students lists? Must all elements be in the plural form or can some be left in singular, even though the meaning is plural? I've seen an expression universities student lists; is it correct? I want to have both KDE and Gnome shells on Ubuntu. Ubuntu only has Gnome installed by default. How do I install KDE? Velocity Maxwell-Boltzmann distribution for dummies I have a volume with $N$ molecules; I need to assign to each particle a velocity vector: $$|\mathbf{v}_{i}|=[v_{x}, v_{y}, v_{z}]^{T}$$ for the $i$-th molecule; the velocities must follow the Maxwell-Boltzmann distribution; here is the question: I know the definition of the M.B. distribution, how do I extract the velocities that I have to assign to each molecule? Basically what I want to know is how to calculate $$v_{x}=?$$ $$v_{y}=?$$ $$v_{z}=?$$ for each molecule. Merge some tab-delimited files (already output) in Debug logs I am trying to add a list into an another nested list before adding values of a list to nested list it is getting the correct value but after adding i am getting (already output) in debug log when i checked.    public List&lt;String&gt; fieldName {get;set;}  public List&lt;List&lt;String&gt;&gt; fileValuesXML; // To get the field values of xml  for(Dom.XMLNode child : SalesforceObjecet.getChildElements()) {                        objecetName = child.getText();             system.debug('Object Name in XML field  ' + objecetName );              for(Dom.XMLNode subchild : child.getChildElements())              {                 system.debug('Sub child ' + subchild.getText() );                 for(Dom.XMLNode subchildfield : subchild.getChildElements())                 {                     system.debug('Sub child Field ' + subchildfield );                     fName  = subchildfield.getText();                     fieldName.add(fName);                     //fileValuesXML.add(fieldName);                     //system.debug('Filed Name in XML field  ' + fieldName);                 }                  system.debug('Filed Name before affing to filevalueXML  ' + fieldName);                                     fileValuesXML.add(fieldName);                 fieldName.clear();                 system.debug('Filed Name after affing to filevalueXML  ' + fileValuesXML);             }    }    I am getting values in debug log as:     Filed Name before affing to filevalueXML  (AccountSource, field3)   Filed Name after affing to filevalueXML  ((), (already output),   (already output)) How should different sized RAM be added to iMac? I have a mid-2011 iMac which came with 4GB RAM (2x2GB). I just bought 8 GB (2x4GB) and added it to get a total of 12GB. The current configuration is as follows:    Given that the RAM is a different size, do I have the optimal locations for the different RAM chips? At the moment what I did was to simply add the 2 new sticks of RAM in the two empty slots.