[行列解析4.4.P19]tr(\(A\overline{A}\)) の実数性とトレース不等式

\begin{aligned}
\mathrm{tr}(A\overline{A})
&=\sum_{i=1}^n(A\overline{A})_{ii}\\
&=\sum_{i=1}^n\sum_{j=1}^n a_{ij}\overline{a}_{ji}.
\end{aligned}

a_{ii}\overline{a}_{ii}=|a_{ii}|^2

a_{ij}\overline{a}_{ji}
+
a_{ji}\overline{a}_{ij}
=
2\operatorname{Re}\left(a_{ij}\overline{a}_{ji}\right)
\in\mathbb R
\qquad (i\neq j)

\mathrm{tr}(A\overline{A})\in\mathbb R

A=
\begin{bmatrix}
0&1\\
-1&0
\end{bmatrix}

A\overline{A}=A^2
=
\begin{bmatrix}
-1&0\\
0&-1
\end{bmatrix}

\mathrm{tr}(A\overline{A})=-2 \lt 0

\begin{aligned}
\mathrm{tr}(AA^*)
&=\sum_{i=1}^n\sum_{j=1}^n
a_{ij}\overline{a_{ij}}\\
&=\sum_{i=1}^n\sum_{j=1}^n|a_{ij}|^2.
\end{aligned}

\mathrm{tr}(A\overline{A})
=
\sum_{i=1}^n|a_{ii}|^2
+
2\sum_{1\leq i \lt j\leq n}
\operatorname{Re}\left(a_{ij}\overline{a}_{ji}\right)

\operatorname{Re}(z\overline{w})
\leq
|z\overline{w}|
=
|z||w|
\leq
\frac{|z|^2+|w|^2}{2}

2\operatorname{Re}\left(a_{ij}\overline{a}_{ji}\right)
\leq
|a_{ij}|^2+|a_{ji}|^2

\begin{aligned}
\mathrm{tr}(A\overline{A})
&=
\sum_{i=1}^n|a_{ii}|^2
+
2\sum_{i\lt j}
\operatorname{Re}\left(a_{ij}\overline{a}_{ji}\right)\\
&\leq
\sum_{i=1}^n|a_{ii}|^2
+
\sum_{i \lt j}
\left(|a_{ij}|^2+|a_{ji}|^2\right)\\
&=
\sum_{i,j}|a_{ij}|^2\\
&=
\mathrm{tr}(AA^*).
\end{aligned}

\boxed{\mathrm{tr}(AA^*)\geq\mathrm{tr}(A\overline{A})}