Find the two principal components of the covariance matrix
$$ \Sigma= \pmatrix{6,~-1\\2,~~~~~3}$$
I have found the eigenvectors as $\lambda_1=4,$ and $\lambda_2=5$ with corresponding eigenvectors $\pmatrix{1\\2}$ and $\pmatrix{1\\1}$.I think if we concatenate the eigenvectors together in descending order we can then get the PCs as
$Y=\pmatrix{1,1\\1,2}\pmatrix{6,~-1\\2,~~~~~3}=\pmatrix{8,2\\10,5}$
is this right? I'm not sure...
Also, if I were to calculate the principal components from the correlation matrix, would the steps be identical?
The two eigenvectors that you obtained are the unit vectors i.e axis of your principal components. The vector with the higher eigen value is the the first principal component i.e. its the direction towards which the variance has been maximised