# How to comment a projection of data in Principal Composant Analysis?

#### AntoineCompagnie

##### New Member
From:

$$A=\begin{bmatrix} 1 & 0 & 0\\ 0 & 0 & 1\\ 0 & 1 & 2\\ 2 & 2 & 1\\ 1 & 0 & 0\\ 2 & 3 & 2\\ \end{bmatrix}$$

I first had to calculate the gravity point g (1,1,1), Y the centered data matrix
$$Y=\begin{bmatrix} -1 & -1 & -1\\ -1 & -1 & 0\\ -1 & 0 & 1\\ 1 & 1 & 0\\ 0 & -1 & -1\\ 1 & 2 & 1\\ \end{bmatrix}$$

and V the covariance matrix with $$V=\frac{1}{n}Y^tY$$
$$V=\begin{bmatrix} 4 & 4 & 0\\ 4 & 8 & 4\\ 0 & 4 & 4\\ \end{bmatrix}$$

There is three eigen vectors but only two eigen vectors non associated with a null eigen value:

$$v_1=\begin{bmatrix} 1\\ 0\\ -1 \end{bmatrix}v_2=\begin{bmatrix} 1\\ 2\\ 1 \end{bmatrix}$$

After normalizing the vectors, projecting Y (the matrix above centralized and normed) on the plane described by the two eigen-vectors, I get the following principal componant analysis:
$$C=\{\frac{C_1}{\sqrt{\lambda_1}},\frac{C_2}{\sqrt{\lambda_2}}\}$$
$$C=\frac{\sqrt{3}}{2} \begin{bmatrix} 1 & -1\\ -1 & -1\\ -1 & 0\\ 1 & 1\\ 1 & -1\\ 0 & 2\\ \end{bmatrix}$$

I then had to display graphically the cloud N'I) of all elements on the factorial plan defined by the two first factorial axes. Furthermore, I have to coment this graphical displaying.