# How to get transformation matrix for Linear Discriminant Analysis from eigen values?

I am trying to implement Linear Discriminant Analysis. I have 2 questions.

A)Can I directly use the matrix with eigen vectors of the product of between scatter matrix inverse and within scatter matrix ( $Sb^{-1} Sw$) as the transformation matrix?

B)Does scaling of eigen vectors by different values affect the LDA output? When I found the eigen values of $Sb^{-1} Sw$ through C programming, the eigen values are the same as what i get by using eig function in Matlab, but the eigen vectors are scaled by different scalars.(Different eigenvectors are scaled by different scalars)

Could some please share their idea on these?

b) Eigenvectors in MATLAB should be normalized to have Euclidean length 1. I'm surprised the same is not true in C. If $\textbf{x}$ is an eigenvector associated with eigenvalue $\lambda$ then so is $c\textbf{x}$ for any constant $c$. This is why it is convention to normalize them, but still its common for the result given by different languages/algorithms to differ by sign (-1).