Finding vectors that direct the discriminatory factorial axis

Question

Let be a set $E$ of $100$ individuals for who the quantitative variables $x_1$ and $x_2$ have been observed. This set is partitioned into $C_1$ and $C_2$ which contains $40$ and $60$ individuals respectively. Each individuals have a weight of $1/100$. We write $g_1$ and $g_2$ the gravity center of $C_1$ and $C_2$ and $V$ the total variance matrix, that is to say the matrix of variable $x_1$ and $x_2$ which have been observed for each of the $100$ individuals. We assume that :

$$g_1=\begin{pmatrix}6\\1 \end{pmatrix}, g_2=\begin{pmatrix}1\\-4 \end{pmatrix}$$

I'm looking for the vector that direct the only discriminatory factorial axis of the following matrix :

$$V= \begin{pmatrix} 5 & 0\\ 0 & 2 \end{pmatrix}$$

I have the following eigenvectors and values :

• $\begin{pmatrix}1\\0\end{pmatrix}$ for $5$
• $\begin{pmatrix}0\\1\end{pmatrix}$ for $2$

Normalizing the vector I get :

• $v_1=\frac{1}{\sqrt{5}}\begin{pmatrix}1\\0\end{pmatrix}$
• $v_2=\frac{1}{\sqrt{2}}\begin{pmatrix}0\\1\end{pmatrix}$

Answer 1

Here is the formula to apply :

\begin{align} \frac{g_2-g_1}{||g_2-g_1||}&= \frac{\begin{pmatrix}1\\-4\end{pmatrix}-\begin{pmatrix}6\\1\end{pmatrix}}{||\begin{pmatrix}1\\-4\end{pmatrix}-\begin{pmatrix}6\\1\end{pmatrix}||}\\ &=\frac{-5\begin{pmatrix}1\\1\end{pmatrix}}{||-5\begin{pmatrix}1\\1\end{pmatrix}||}\\ &=-\frac{1}{2}\begin{pmatrix}1\\1\end{pmatrix} \end{align}