distinct eigenvalues

Question

When we say that we have ordered distinct eigenvalues. Does that mean that the the eigenvalues are decreasing $\lambda_1 \geq \cdots \geq \lambda_p$ or strictly decreasing $\lambda_1 > \cdots > \lambda_p$ ?

Answer 1

If the $\lambda_i$ represent all of the eigenvalues, then the first notation is correct, since you will have equality where there are repeated eigenvalues and the order is not strict.

But if the $\lambda_i$ represent only the distinct eigenvalues, then the second notation is correct, since distinct means that the repeats are not included and the order is strict.

For example, if the eigenvalues are $3, 3, 3, 2, 1, 1,$ then you would order the distinct eigenvalues as $3>2>1$. You could order all of the eigenvalues as $3\geq 3\geq 3\geq 2\geq 1\geq 1$.