Eigenvalues of a $3\times 3$ symmetric matrix Given a $3\times 3$ symmetric matrix  \begin{equation}  M= \begin{pmatrix} A &amp; B &amp; C \\ B &amp; D &amp; E \\  C &amp; E &amp; F\\  \end{pmatrix}, \end{equation}  how do I find the eigenvalues? Brute-forcing it by looking at the roots of the characteristic polynomial seems overwhelming.  Are there any properties of symmetric matrices that might simplify the problem or am I overlooking something?
General Information about Eigenvalues for an 3x3 symmetric matrix How do we find general information about the eigenvalues of an arbitrary 3x3 symmetric matrix without resorting to explicitly computing the solutions to the cubic characteristic equation?:  \begin{bmatrix}     a   &amp; b &amp; c  \\     b    &amp; d &amp; e \\     c &amp; e &amp;f \end{bmatrix}  (In this case the sum and product of the eigenvalues, and the sum of the reciprocals of the eigenvalues.) Since diagonalization preserves the trace and determinant of matrices, would it be best to just find the corresponding diagonal matrix and read off the sum and product of the eigenvalues from there? The problem is I can't think of a way to diagonalize a matrix without explicitly computing the eigenvalues.
How can I modify the size of swap with LVM partitions? I'm trying to add/modify a swap partition on my LVM-formatted hard disk by using GParted from a LiveCD.  I only see one big partition taking up the whole disk.    Is there a hidden swap?    Does an LVM-formatted system even need a swap?  I'm using Ubuntu 12.10. A problem with my reasoning in a problem about combinations I'm trying to install wifi drivers for a particular mini wifi card i bought. The output of lsusb is:  0bda:f179 Realtek Semiconductor Corp   I've searched everywhere for drivers, but could not find anything. Any help?  Thanks A 64-bit double can represent integer +/- 253 exactly. Given this fact, I choose to use a double type as a single type for all my types, since my largest integer is an unsigned 32-bit number. But now I have to print these pseudo integers, but the problem is they are also mixed in with actual doubles. So how do I print these doubles nicely in Java? I have tried String.format(&quot;%f&quot;, value), which is close, except I get a lot of trailing zeros for small values. Here's an example output of of %f  232.00000000 0.18000000000 1237875192.0 4.5800000000 0.00000000 1.23450000  What I want is:  232 0.18 1237875192 4.58 0 1.2345  Sure I can write a function to trim those zeros, but that's lot of performance loss due to string manipulation. Can I do better with other format code?  The answers by Tom E. and Jeremy S. are unacceptable as they both arbitrarily rounds to two decimal places. Please understand the problem before answering.  Please note that String.format(format, args...) is locale-dependent (see answers below). Can I travel to the USA while working remotely for my non-US employer? I would like to travel in the USA for a period of 3 - 6 months. But I want to spend some time working for my employer (remote work) and getting paid by the employer. Is this possible under a tourist visa (B1/B2)?  I hope this is OK since I will not be getting paid by a US employer. I am in the IT trade so all that I will be needing is my own laptop and a connection to the internet. The money I earn would actually help me pay for the expenses I get during the holiday. Prevent Mac Terminal from reverting to default theme I can't find any way to change the Terminal theme and have that theme applied permanently.  I want to change to another darker theme other than the default 'basic' theme, and (most importantly) have it remain as the theme every time I open Terminal. Q&A Identification of seldom man-made physical objects not being piece of art? Is SE Q&amp;A format relevant for "identification of seldom physical objects observed in real world"?  Questions like "I do not know function of this object (picture) - which purpose does it have?"  In many cases I suppose these question could have civil engineering or handcraft i.e. cultural/historical background. In second case "SE history" might be okay. For first case, I do not know. Multinomial logistic regression analysis or binary logistic regression Should I round my uncertainty to nearest hundred? I was doing some exercises on error propagation when I came across this problem:  $4\pi^2/(0,034 \pm 0,004 \space s^2/cm)$  I calculated my uncertainty to be $\Delta$ = 136 $cm/s^2$  And so the hole thing gives me: $(1200 \pm 136) \space cm/s^2$.  Now going into my question I rounded $4\pi^2/0,034 = 1161.12993$ to $1200$ which makes sense since my uncertainty is 136 but in the solution to this problem $136$ is rounded to $100$ and that's not making much sense to me.  So should I always round my uncertainty to the nearest hundred (assuming my uncertainty has 3 digits)? Why does a window become a mirror at night? In day, when you look in the room through the window out, you can clearly see what happens outside. At night when it's dark outside but there's light inside you can look in the window but it becomes a mirror.  Why? How to see an embedded image's dpi (ppi) in a pdf in Acrobat Pro? The following situation baffled me slightly: after preparing a pdf in a different program I wanted to make sure that an embeded image in the pdf file had the right dpi, yet I found no way in adobe acrobat pro to inspect the file/image dpi.   Can that be, or am I missing where this feature is hidden? How to to calculate the hash of an unordered set Suppose I have a set of elements, with known hash (e.g. SHA-2). How can I calculate the hash of the set? With it I mean an unordered set, so the order of elements is undefined and shall not play any role in determining the hash of the set.  In theory any commutative operation applied to the hashes should be fine, but I have the strong feeling that XORing all of them together is quite unsafe. An attacker could add an element to the set to produce a hash collision, although finding an element with that wanted hash is as hard as breaking the hash of the whole set.  Anything better solution than the trivial one of sorting the elements into a list, calculating their hashes and then calculating the hash of the hash list?  I think Merkle trees can't help here if they are based on some sort of ordering of the elements. While modeling an Audi R8 I have the following problem. The thing I want to create is seen in this image:    I want to create a sharp crease (outlined in red) which starts around the first exhaust and then goes to the second.  Is there an efficient way to do this (I also use the Subdivision Surface Modifier for this model). Why use a prime number in hashCode? Obi-Wan left Anakin for dead, and one movie later (in A New Hope) he was saying Anakin had been killed by Darth Vader.  What exactly did Obi-Wan know about Anakin and Darth Vader before A New Hope started? Omitting 'that' in this sentence To turn off my system, I can use one of these three commands:  sudo halt   Or:  sudo poweroff   Or:  sudo shutdown -h now   What are the differences between these commands, or are they all just aliases of each other? PHP PDOException: "SQLSTATE[HY093]: Invalid parameter number" How to match * with hidden files inside a directory How to match the hidden files inside the given directories   for example   If I give the below command it's not giving the result of the hidden files,     du -b maybehere*/*   how to achieve this simple using a single command instead of using   du -b maybehere*/.* maybehere*/*   as I need to type maybehere twice. Grid search vs. random search for hyperparameter tuning I am interested in comparing the computational cost of grid search vs. random search algorithm for hyperparameter tuning. Is random search computationally less expensive then grid search? Is there an article/paper that talks about this topic? Thank you,