When Are Eigenvalues Distinct?

"Distinct" numbers just means different numbers. If a and b are eigen values of operator T and then they are "distinct" eigenvalues. If they happen to be 0 and 1, then, since they are different, they are "distinct".

What makes an eigenvalue distinct?

The distinct eigenvalues of A are 0,1,2. When eigenvalues are not distinct, it means that an eigenvalue appears more than once as a root of the characteristic polynomial. In geometric terms, it means that there are multiple linearly independent vectors that the matrix scales by the same constant.

Do eigenvalues have to be distinct?

A matrix does not necessarily have distinct eigenvalues (although almost all do), and a matrix does not necessarily have a single eigenvalue with multipicity n. In fact, given any set of n values, you can construct a matrix with those values as eigenvalues (indeed just take the corresponding diagonal matrix).