Here's the matrix $A$: \begin{pmatrix} -1 &amp; 1 &amp; 2\\ 2 &amp; -2 &amp; 3\\ -1 &amp; 1 &amp; -2 \end{pmatrix} How do I solve the matrix equation $XA = X - A$?
How does one solve the matrix equation $AX+XB=C$ for $X$? It doesn't seem too difficult. I tried many times but failed.  I'm an adult student... I am now vexed about Gilbert Strang - An Introduction to Linear Algebra. I don't even understand a single word in Wikipedia: . If you have ever use some nice workable materials or lecture notes? You can generously upload and share the links of the lecture notes and assignments. Different subjects/ topics are welcome, as long as you deem they are nice and workable.   The problem origins from a system of diff equation, using undetermined coefficients (matrix) to find the particular solution. Try $y_p=X\begin{pmatrix}   e^{\alpha t} \\ e^{\beta t} \end{pmatrix}$  $\dot{y}+Ay=C\begin{pmatrix} e^{\alpha t} \\ e^{\beta t}\end{pmatrix}$   $\dot{y_p}=X\begin{pmatrix} \alpha &amp; 0 \\ 0 &amp; \beta \end{pmatrix} \begin{pmatrix}e^{\alpha t} \\ e^{\beta t}\end{pmatrix}$  substitute $\dot{y_p}$ and $y_p$ into the original differential equation..  $X\begin{pmatrix} \alpha &amp; 0 \\ 0 &amp; \beta \end{pmatrix}+AX=C$
Problem aligning overlapping corner points in Illustrator I am drawing some overlapping paths in Illustrator. However, some anchor points of different paths do not align correctly, they create some rugged corners. I tried to manually adjust the points with the direct selection tool but it is difficult to handle the corners.  Can anyone help me to fix the corners in the red circles? I want to compile my program with the latest version of gcc.  Ubuntu 14.04 comes with gcc 4.8.2, however there's 4.9.0 available, moreover, I see that it is available as a package: . I tried to install it  sudo apt-get install gcc-4.9   but it says  Reading package lists... Done Building dependency tree        Reading state information... Done Note, selecting 'gcc-4.9-base' for regex 'gcc-4.9' gcc-4.9-base is already the newest version. 0 upgraded, 0 newly installed, 0 to remove and 0 not upgraded.   Looks like it is already installed, just not as the default one? How do I utilize it to build my program? Securely erasing all data from a hard drive I am about to sell my old desktop PC and I am cautious about some of my sensitive information being available to the purchaser, even after reformatting the hard-drive, using data recovery software.  How can I securely wipe the hard drive so that the data on it cannot be recovered?  Although I specifically want help with my Windows PC, it wouldn't hurt if there were suggestions for Macs as well. No sound after upgrade to 18.10, only a dummy device is shown After I updated to Ubuntu 18.10 my sound device isn't recognized anymore. Only a dummy device is shown.  I have an onboard Intel device:  lspci -nnk | grep -A2 Audio 00:1b.0 Audio device [0403]: Intel Corporation 7 Series/C216 Chipset Family High Definition Audio Controller [8086:1e20] (rev 04)     Subsystem: Intel Corporation 7 Series/C216 Chipset Family High Definition Audio Controller [8086:2036]     Kernel driver in use: snd_hda_intel     Kernel modules: snd_hda_intel   The command pacmd list-cards does not show any devices:  :~$ pacmd list-cards 0 card(s) available.   This message in /var/log/syslog might have to do something with the problem:  Oct 19 21:02:06 MyComputerName pulseaudio[14331]: E: [pulseaudio]  backend-ofono.c: Failed to register as a handsfree audio agent with  ofono: org.freedesktop.DBus.Error.ServiceUnknown: The name org.ofono  was not provided by any .service files   Does anybody have the same issue? How can I fix this? Today I was informed by VFS that my tourist visa for UAE got approved. I'm travelling to UAE on 16th December 2014 and returning on 7th Jan 2015. But when I checked the "valid until" date on the visa it states 30 December 2014.    Does this date mean I should return before December 30th? Should I get it corrected from VFS? When you're logged in to a Stack Exchange account, it's possible to add and remove logins (associated OpenID/OAuth accounts) without performing any form of reauthentication.  This is not a full vulnerability in itself, but it provides an extremely easy way for an attacker who's found a temporary XSS vulnerability to turn that into a permanent account hijack. An XSS exploit on its own may only provide a minute of indirect access to a user's account. But with this login configuration, the attacker could use that minute to automatically:   Add a new attacker-controlled login to the account. For example:   The XSS JavaScript would generate a random identifier, and use that to construct a login URL of the form http://hacker.example/openid/1234567. The hacker.example server would automatically authenticate the first OpenID login request made with a given identifier, allowing it to be added to the account without user interaction. All subsequent login attempts would be automatically rejected for the user. The hacker.example server would notify the operator with the account name and the generated identifier, and would allow them to login to it.  Remove all existing logins from the account. Log the user out of their account, with no way to get back in.   If this exploit were used to hijack moderator and staff accounts it could makes it more difficult to quickly respond to an attack, creating a greater opportunity for extended abuse.  This is a broader instance of the same fundamental vulnerability which was previously-reported as . That instance is a little more straightforward to fix: just ask for the password again when changing it. However, safe reauthentication with OpenID/OAuth logins is more complicated. Most users' OpenID providers will be configured to automatically re-authenticate upon request, so there would be minimal security gain from requesting that.  Instead or as well, my suggestion is that for users that have a verified email address, require an emailed confirmation link to be clicked before committing any login changes.    I'm not a real security expert, but I've just found that fellow Stack Overflow user , who is, previously described this vulnerability in his 2012 blog post :     If an attacker can add additional login accounts then they can create a persistent mechanism to get back into that Stack Overflow account at any time without needing to hijack the session. Now of course the victim can always go into their account settings and identify the presence of the new login, but it’s a question of how much damage is done before then.      How would an attacker mitigate the risk of having their newly added account removed? Just remove the victim’s original account(s) instead.      [...]      Finally, this is also a lesson for those considering using third party authentication providers. Yes, it’s great that you can farm off this work to someone who has a solid implementation, but you still have responsibility for securing the authenticated session on your site. If you choose not to, be aware of the risks because certainly we’ve seen them exploited before. I have a doubt on something. I have a theorem that says that if $R$ is a ring, then $R/I$ is a domain iff $I$ is prime. Since $X^2-1$ is note prime, $\mathbb Z[X]/(X^2-1)$ is not a domain. But, $X^2-1=(X-1)(X+1)$, and since $X-1$ and $X+1$ are prime in $\mathbb Z[X]$, there are coprime, and thus, by the chinese remainder $$\mathbb Z[X]/(X^2-1)\cong \mathbb Z[X]/(X-1)\times \mathbb Z[X]/(X+1),$$ since $\mathbb Z[X]/(X-1)\times \mathbb Z[X]/(X+1)$ is a domain, then so is $\mathbb Z[X/(X^2-1)$. What is the mistakes here ? The process substitution output is out of the order How can I delete my account? What is butterfly lighting, and when do I use it? I often hear the term 'Butterfly Lighting' used when discussing high-fashion and glamor photography. How do I set up Butterfly Lighting, and when do I use it? Finding multivariable limits for the function $\frac{3x^2y}{x^2+y^2}$ How do you change from icon to list view in 20.4 C structure initialization? string.format format string containing { Question on the coefficients of $(1+x+x^2+x^3+x^4)^{496}$ Consider the expansion $$(1+x+x^2+x^3+x^4)^{496} = a_0+a_1x+\cdots+a_{1984}x^{1984}.$$   $\quad$ (a) Determine the greatest common divisor of the coefficients $a_3,a_8,a_{13},\ldots,a_{1983}$.      $\quad$ (b) Prove that $10^{340} &lt; a_{992} &lt; 10^{347}$.   Is there an easier way to solve this and is there a formula for the multisection for just the sum of the coefficients and not including $x^k$?  I thought of using the Multisection formula to prove (a). That is, $$\sum_{k \equiv r \pmod{m}}a_kx^k = \dfrac{1}{m} \sum_{s=0}^{m-1} \epsilon^{-rs} f(\epsilon^s x)$$ where $\epsilon$ is a primitive $m$th root of unity and $\displaystyle f(x) = (1+x+x^2+x^3+x^4)^{496}$.  Thus we have $r = 3, m = 5$ and so \begin{align*}\sum_{k \equiv 3 \pmod{5}}a_kx^k &amp;= \frac{1}{5} \sum_{s=0}^{4} w^{-3s} (1+w^s x+(w^{s}x)^2+(w^{s} x)^3+(w^{s}x)^4)^{496}\\&amp;=\dfrac{1}{5}\sum_{s=0}^4 w^{-3s}\left(\dfrac{(w^s x)^{5}-1}{w^s x-1}\right)^{496}\\&amp;= \dfrac{1}{5}\sum_{s=0}^4 w^{-3s} \left(\dfrac{x^{5}-1}{w^s x-1}\right)^{496}.\end{align*} Why doesn't a block rotate due to friction? Let $f(x)$ and $g(x)$ be two continuous functions on $[0,1]$ and  $$\int_{0}^{1}f(x) dx= \int_{0}^{1}g(x)dx = 1$$   Show that there exist  $[a,b]\subset [0,1]$, such that  $$\int_{a}^{b}f(x) dx= \int_{a}^{b}g(x)dx = \frac{1}{2} $$  The question can be solved by considering the fundamental group of $S^1$, now I am wondering if we can solve it by real-analysis.  Here is the topological solution:  Assume that for any $[a,b]\subset[0,1]$, we always have  $$(\int_{a}^{b}f(x) dx\neq \frac{1}{2}) \quad \vee \quad(\int_{a}^{b}g(x)dx \neq \frac{1}{2}) $$ Consider mapping $$\phi:D\rightarrow\mathbb{E}^2\backslash\{(\frac{1}{2},\frac{1}{2})\},\,\,(x,y)\mapsto(\int_{y}^{x}f(t) dt,\int_{y}^{x}g(t) dt)$$ where $D=\{(x,y)|0\leq x\leq y\leq 1\}$.  Let $a$ be path from $(0,0)$ to $(0,1) $in $D$ and $b$ be path from $(0,1)$ to $(1,1) $in $D$, then $ab$ is a path from $(0,0)$ to $(1,1) $in $D$ and $$\phi\circ (ab):[0,1]\rightarrow\mathbb{E}^2\backslash\{(\frac{1}{2},\frac{1}{2})\} $$ Notice that $$(\phi\circ (ab))(0)=(\phi\circ (ab))(1)=(0,0)$$  so $\phi\circ (ab)$ is a loop based on $(0,0)$ in $\mathbb{E}^2\backslash\{(\frac{1}{2},\frac{1}{2})\}$.  For any $t\in [0,\frac{1}{2}]$, it's not hard to get that $$(\phi\circ (ab))(t+\frac{1}{2})=(1,1)-(\phi\circ (ab))(t)$$ is equivalent to $$(\phi\circ (ab))(t+\frac{1}{2})-(\frac{1}{2},\frac{1}{2})=-((\phi\circ (ab))(t)-(\frac{1}{2},\frac{1}{2}))$$  Define retraction  $$r:\mathbb{E}^2\backslash\{(\frac{1}{2},\frac{1}{2})\}\rightarrow S^1,\quad (x,y)\rightarrow\ \frac{(x,y)-(\frac{1}{2},\frac{1}{2})}{||(x,y)-(\frac{1}{2},\frac{1}{2})||}$$  Then $r\circ \phi\circ (ab)$ is a loop on $S^1$, such that $$(r\circ \phi\circ (ab))(t+\frac{1}{2})=-(r\circ \phi\circ (ab))(t),\quad\forall t\in [0,\frac{1}{2}]$$  So $&lt;r\circ \phi\circ (ab)&gt;$ is not trivial in $\pi_1(S^1)$.   However, $ab$ is path homotopic to $c$ in $D$, where $c$ is the path between $(0,0)$ and $(1,1)$ in $D$. In this case, $r\circ \phi\circ (ab)$ is a point-path in $S^1$ by $$(\phi\circ c)(t)=\phi(t,t)\equiv (0,0)$$  which leads to contradiction. Deleting self-answered question gets "Delete this answered question" warning I'm a new blender user, but have used other modelling programs before.   I've downloaded a  and unzipped it.   When I import the obj into blender, it is my understanding (from here: ) that the mtl should just import itself with it. However, when I import the obj, I just get a grey model with no texture. Is it actually importing the mtl but I'm not seeing it? Also for the record, I have changed no settings in blender.  Picture for reference: How do I quickly rename a MySQL database (change schema name)? The MySQL manual at  covers this.  Usually I just dump the database and reimport it with a new name. This is not an option for very big databases. Apparently RENAME {DATABASE | SCHEMA} db_name TO new_db_name; .  This needs to work with , which stores things very differently than .