The question is to perform Principal Component Analysis $($PCA$)$ on a given dataset. The dataset consists of $5$ data instances with $3$ attributes. The following steps are required:

$1.$ Calculate the multivariate mean $($vector of mean for each attribute$).$

$2.$ Center the data matrix.

$3.$ Calculate the covariance matrix of the centered data $($using unbiased estimation$).$

$4.$ Calculate the eigenvalues of the covariance matrix.

$5.$ Calculate the principal components and transformation matrix,

a$.$showing the correspondence between eigenvalues and principal components.

$6.$ Transform the original data matrix using the PCA transformation.

$7.$ etermine the total number of principal components, the variance explained by the first principal component, and the number of principal components that should be kept.

Instance Attribute $1$ Attribute $2$ Attribute $3$

A$1147$

A$21131$

A$3228$