\listfiles
\documentclass{jarticle}
\usepackage{kpfonts}

\def\trmatrix#1{%
  \sideset{^t\!\!}{}{\mathop{\begin{pmatrix}#1\end{pmatrix}}}
}

\begin{document}

\begin{align*}
&\iint\limits_{\text{region } x,y}\\
&\sqrt{x_{n}^2}=\sqrt{x_{n+1}^2}
\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt x}}}}}}}\\
&\trmatrix{a_{11} & a_{12} & a_{13} \\
  a_{21} & a_{22} & a_{23}\\
  a_{31} & a_{32} & a_{33}}\\
&(P \boldsymbol{y},AP \boldsymbol{y})={}^t(P \boldsymbol{y})AP \boldsymbol{y}\\
&=(y_1,\cdots,y_n) \left( \begin{array}{ccc}
\lambda_1 & & O \\
& \ddots & \\
O & & \lambda_n 
\end{array}
\right) 
\left( \begin{array}{c}
y_1 \\ \vdots \\ y_n
\end{array}\right)\\
&=\lambda_1 y_1^2+\cdots+\lambda_n y_n^2 \leq 1
\end{align*}

\end{document}
