minor bugfixes

nodes/abstract-algebra/groups/group.html

@@ -52,7 +52,7 @@ <h2 class="no-deps"> Dependencies:
 \[ g^k = \begin{cases}
 g*g*\ldots*g \; (k \textrm{ times}) &amp; \textrm{if } k &gt; 0
 \\ e &amp; \textrm{if } k = 0
-\\ g^{-1}*g^{-1}*\ldots*g^{-1} \; (k \textrm{ times}) &amp; \textrm{if } k &lt; 0
+\\ g^{-1}*g^{-1}*\ldots*g^{-1} \; (-k \textrm{ times}) &amp; \textrm{if } k &lt; 0
 \end{cases} \]</p>
 <p>$\operatorname{order}(g)$ is the smallest positive integer $k$ such that $g^k = e$.
 If such a $k$ does not exist, $\operatorname{order}(g) = \infty$.</p>

nodes/linear-algebra/eigenvectors/real-matrix-with-real-eigenvalue-has-real-eigenvectors.html

@@ -54,15 +54,23 @@ <h2> Dependencies:
 </ol>
 
 <div class="horizontal-rule"></div>
-<p>Let $A$ be a real matrix. Let $A$ have a real eigenvalue $\lambda$ and a corresponding complex eigenvector $u$.
+<p><span class="invisible">
+$\newcommand{\ubar}{\overline{u}}$
+</span>
+Let $A$ be a real matrix. Let $A$ have a real eigenvalue $\lambda$ and a corresponding complex eigenvector $u$.
 Then a real eigenvector of $A$ exists corresponding to the eigenvalue $\lambda$.</p>
 <h2>Proof</h2>
-<p>\[ Au = \lambda u
-\implies \overline{A}\,\overline{u} = \overline{\lambda}\,\overline{u}
-\implies A\overline{u} = \lambda\overline{u}
-\implies A(u + \overline{u}) = \lambda(u + \overline{u}) \]</p>
-<p>Therefore, $v = u + \overline{u}$ is an eigenvector of $A$ corresponding to $\lambda$.
-Also, $v = u + \overline{u} = 2\operatorname{Re}(u)$ is real.</p>
+<p>\begin{align}
+&amp; Au = \lambda u
+\implies \overline{A}\,\ubar = \overline{\lambda}\,\ubar
+\implies A\ubar = \lambda\ubar
+\\ &amp;\implies A(u + \ubar) = \lambda(u + \ubar)
+    \textrm{ and } A(i(u - \ubar)) = \lambda(i(u - \ubar))
+\end{align}</p>
+<p>$u + \ubar = 2\operatorname{Re}(u)$ and $i(u - \ubar) = -2\operatorname{Im}(u)$ are real.
+At least one of them must be non-zero, since $u + \ubar = u - \ubar = 0 \implies u = 0$,
+which contradicts the fact that $u$ is an eigenvector of $A$.
+Hence, one of $u + \ubar$ and $i(u - \ubar)$ is a real eigenvector of $A$ corresponding to $\lambda$.</p>
 
 <div class="horizontal-rule"></div>
 <h2> Dependency for: </h2>