The calculation answering the question in the title is not that difficult, but I needed it recently, and so I thought I would add it to my blog.

Suppose we have a covariance matrix

By definition, the eigenvalues are the solutions to the characteristic equation . From the quadratic formula we obtain

Repeated eigenvalues occur precisely when the discriminant (the expression under the square root sign) is . Thus repeated eigenvalues occur when ,

or, simplifying,

Thus has repeated eigenvalues precisely when the two random variables are uncorrelated and have the same variance; i.e., and .

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Your conclusion is true in general for symmetric matrices from the fact that any symmetric matrix is diagonalizable. Hence, . If all the eigenvalues are equal to , then .

Thanks for that generalization, Marvis – not to mention a simpler proof than the one in my post.