diff --git a/src/op/system/two.tex b/src/op/system/two.tex
index 03f5029..95d9af6 100644
--- a/src/op/system/two.tex
+++ b/src/op/system/two.tex
@@ -1,4 +1,4 @@
-\section{On the Subject of 2×2 Matrices}
+\section{On 2×2 Matrices}
 \label{section:op-sys-2by2}
 
 \begin{lemma}
@@ -194,7 +194,7 @@
         \item $\left[\begin{smallmatrix} P & A \\ A^* & Q \end{smallmatrix}\right] \ge 0$.
         \item For each $\xi, \eta \in H$, $|\dpn{A\xi, \eta}{H}| \le \dpn{Q\xi, \xi}{H}^{1/2}\dpn{P\eta, \eta}{H}^{1/2}$.
         \item $P \ge 0$, $A(H) \subset P^{1/2}(H)$, and $Q - (P^{-1/2}A)^*(P^{-1/2}A) \ge 0$.
-        \item $Q \ge 0$, $A^*(H) \subset Q^{1/2}(H)$, and $P - (Q^{-1/2}A)^*(Q^{-1/2}A) \ge 0$.
+        \item $Q \ge 0$, $A^*(H) \subset Q^{1/2}(H)$, and $P - (Q^{-1/2}A^*)^*(Q^{-1/2}A^*) \ge 0$.
     \end{enumerate}
 \end{proposition}
 \begin{proof}[Proof, {{\cite[Exercise 3.2]{Paulsen}}}. ]
@@ -267,7 +267,7 @@
     so
 
     \[
-    0 \le Q - (P^{-1/2}A)^*(P^{1/2}A) \le Q - A^*(P+\eps)^{-1}A
+    0 \le Q - (P^{-1/2}A)^*(P^{-1/2}A) \le Q - A^*(P+\eps)^{-1}A
     \]
 
     and $\left[\begin{smallmatrix} P + \eps & A \\ A^* & Q \end{smallmatrix}\right] \ge 0$ by (P2) of \autoref{proposition:schur-complement-positive}. As this holds for all $\eps > 0$, $\left[\begin{smallmatrix} P& A \\ A^* & Q \end{smallmatrix}\right] \ge 0$.
@@ -310,8 +310,8 @@
 
     and
     \begin{align*}
-        \dpn{A + A^*\xi, \xi}{H} &= 2\text{Re}\dpn{A + A^*\xi, \xi}{H} \\
-        |\dpn{A + A^*\xi, \xi}{H}| &\le 2 \dpn{P\xi, \xi}{H}^{1/2}\dpn{Q\xi, \xi}{H}^{1/2} \\
+        \dpn{(A + A^*)\xi, \xi}{H} &= 2\text{Re}\dpn{A + A^*\xi, \xi}{H} \\
+        |\dpn{(A + A^*)\xi, \xi}{H}| &\le 2 \dpn{P\xi, \xi}{H}^{1/2}\dpn{Q\xi, \xi}{H}^{1/2} \\
         \dpn{(P + A + A^* + Q)\xi, \xi}{H} &\le \dpn{P\xi, \xi}{H} + 2 \dpn{P\xi, \xi}{H}^{1/2}\dpn{Q\xi, \xi}{H}^{1/2} + \dpn{Q\xi, \xi}{H} \\
         &= (\dpn{P\xi, \xi}{H}^{1/2} + \dpn{Q\xi, \xi}{H}^{1/2})^2 \\
     \end{align*}
@@ -339,7 +339,7 @@
     In particular, for $\xi = -A\eta/\norm{P}_{B(H)}$, 
     \begin{align*}
         0 &\le \frac{\dpn{PA\eta, A\eta}{H}}{\norm{P}_{B(H)}^2} + \dpn{P\eta, \eta}{H} - \frac{\dpn{A\eta, A\eta}{H}}{\norm{P}_{B(H)}} - \frac{\dpn{A^*A\eta, \eta}{H}}{\norm{P}_{B(H)}} \\
-        &= \frac{\dpn{PA\eta, A\eta}{H}}{\norm{P}_{B(H)}} + \norm{P}_{B(H)}\dpn{P\eta, \eta}{H} - 2\norm{A\eta}_H^2  \\
+        0 &\le \frac{\dpn{PA\eta, A\eta}{H}}{\norm{P}_{B(H)}} + \norm{P}_{B(H)}\dpn{P\eta, \eta}{H} - 2\norm{A\eta}_H^2  \\
         &\le\norm{A\eta}_H^2 + \norm{P}_{B(H)}\dpn{P\eta, \eta}{H} - 2\norm{A\eta}_H^2 \\
         &= \norm{P}_{B(H)}\dpn{P\eta, \eta}{H} - \dpn{A^*A\eta, \eta}{H}
     \end{align*}