eqs2

$\displaystyle \bar{x}_{j}$	$\textstyle =$	$\displaystyle {1\over m}\sum_{i=1}^mx_{ij}$
$\displaystyle \sigma_{x_j}^2$	$\textstyle =$	$\displaystyle {1\over m}\sum_{i=1}^m (x_{ij}-\bar{x}_j)^2$	(2)

$\displaystyle \bar{z}_{j}$	$\textstyle =$	$\displaystyle {1\over m}\sum_{i=1}^mz_{ij}=0$
$\displaystyle \sigma_{z_j}^2$	$\textstyle =$	$\displaystyle {1\over m}\sum_{i=1}^m (z_{ij})^2={1\over m}\sum_{i=1}^m\left({x_{ij} - \bar{x}_j\over \sigma_{x_j}}\right)^2=1$	(3)

$\displaystyle Z= \left( \begin{array}{cccc} z_{11} & z_{12} & \ldots & z_{1n} \... ...olumn{4}{c}{\dotfill}\\ z_{m1} & z_{m2} & \ldots & z_{mn} \end{array}\right)$ (4)

$\displaystyle R={1\over m}Z^{T}Z$	$\textstyle =$	$\displaystyle {1\over m} \left( \begin{array}{cccc} z_{11} & z_{21} & \ldots & ... ...icolumn{4}{c}{\dotfill}\\ z_{m1} & z_{m2} & \ldots & z_{mn} \end{array}\right)$
	$\textstyle =$	$\displaystyle {1\over m}\left( \begin{array}{cccc} \sum_{i=1}^{m} z_{i1}^2 & \s... ...um_{i=1}^{m} z_{in}z_{i2} & \ldots & \sum_{i=1}^{m} z_{in}^2 \end{array}\right)$	(5)

$\displaystyle \rho_{jk} ={1/m\sum_{i=1}^m(x_{ij}-\bar{x}_j)(x_{ik}-\bar{x}_k)\over \sqrt{\sum_{i=1}^m(x_{ij}-\bar{x}_j)^2/m\sum_{i=1}^m(x_{ik}-\bar{x}_k)^2}/m}$ (6)

$\displaystyle u_{i}=a_1z_{i1}+a_2z_{i2}+...+a_nz_{in}=\sum_{j=1}^{n}a_jz_{ij}$ (7)

$\displaystyle S$	$\textstyle =$	$\displaystyle {1\over m}\sum_{i=1}^m(u_{i})^2$
	$\textstyle =$	$\displaystyle {1\over m}\sum_{i=1}^m(\sum_{j=1}^{n}a_jz_{ij})^2$
	$\textstyle =$	$\displaystyle {1\over m}\sum_{i=1}^m(a_1^2z^2_{i1}+a_2^2z^2_{i2}+... +2a_1a_2z_{i1}z_{i2}+a_1a_3z_{i1}z_{i3}+..)$
	$\textstyle =$	$\displaystyle a_1^2+a_2^2+...+2\rho_{12}a_1a_2+...$	(8)

$\displaystyle f(a_1,a_2,...a_n,\lambda) = a_1^2+a_2^2+...a_n^2+2\rho_{12}a_1a_2+2\rho_{13}a_1a_3+.... - \lambda(a_1^2+a_2^2+...a_n^2-1)$ (10)

$\displaystyle 2a_1+2\rho_{12}a_2+2\rho_{13}a_3+.... - 2\lambda a_1$	$\textstyle =$	$\displaystyle 0$
$\displaystyle 2\rho_{21}a_1+2a_2+2\rho_{23}a_3+.... - 2\lambda a_2$	$\textstyle =$	$\displaystyle 0$
$\displaystyle .....$
$\displaystyle 2\rho_{n1}a_1+2\rho_{n2}a_2 + .....2a_n - 2\lambda a_n$	$\textstyle =$	$\displaystyle 0$	(12)

$\displaystyle \left( \begin{array}{ccccc} 1 & \rho_{12} & \rho_{13} & \ldots &... ...lumn{1}{c}{\dotfill}\\ a_k\\ \multicolumn{1}{c}{\dotfill} \end{array}\right)$ (13)

$\displaystyle a_1^2+\rho_{12}a_1a_2+\rho_{13}a_1a_3+....$	$\textstyle =$	$\displaystyle \lambda a_1^2$
$\displaystyle \rho_{21}a_2a_1+a_2^2+\rho_{23}a_2a_3+....$	$\textstyle =$	$\displaystyle \lambda a_2^2$
$\displaystyle .....$
$\displaystyle \rho_{n1}a_na_1+\rho_{n2}a_na_2 + .....a_n^2$	$\textstyle =$	$\displaystyle \lambda a_n^2$	(14)