degenerate eigenvalues

Question

I have a problem in understanding the exact meaning of degenerate eigenvalue. I have some database and I calculate the covariance matrix among it. the obtained eigenvalues are same ( all of them =5000) and the obtained eigenvectors are different each other and orthogonal. what does that tell? I know in this case covariance matrix is symmetric and diagonal. eigenspace forms a sphere. But is there any advantage of this eigenspace ( any special property)?

Answer 1

Let your covariance matrix be $X^TX$, where $X$ is the data matrix. Put your eigenvectors together to form an invertible matrix $P$. Your computation result shows that $X^TXP=5000P$, i.e. $X^TX=5000I$.

Since every set of $n$ linearly independent vectors (be they orthogonal or not) form an eigenbasis of a scalar matrix in $\mathbb{R}^n$, I don't think there is anything special about the eigenbasis you obtained. What is special, however, is the data matrix itself. The fact that $X^TX=5000I$ indicates that, without a change of basis, the random variables in question are already uncorrelated and have identical variances ($5000$). If these random variables are jointly normal, we can further conclude that they are mutually independent.