eigenvalues – How to calculate the Covariance and means value form eigenvector matrix

I am trying to solve the 6 exercise from the below eBook:

Fukunaga, Keinosuke. – Introduction to Statistical Pattern Recognition-Elsevier-Academic Press (1999)

as you can see below in this exercise wen need to find the Covariance and mean value from the eigenvector matrix:

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So i have created some python codes for computer problem of this book here:

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So if possible, i like to find out am i right about the way of finding the covariance and mean value from eigenvector by building the x value form the normal distribution and then calculating the Covariance and mean vale for the achieved X Vector, or not?

Thanks.