On the number of linearly independent eigenvectors

Question

Is number of linearly independent eigenvector same as number of distinct eigenvalues of a matrix?

I have seen many of my friends using this shortcut to find out number of linearly independent eigen vectors, but does this hold true in each and every case?

Answer 1

Eigenvectors for different eigenvalues are independent. But eigenvectors for the same eigenvalue can be independent also. This happens when the dimension of the eigenspace is greater than one.

So, we can say that there are at least as many independent eigenvectors as distinct eigenvalues.

Answer 2

It's true only when all eigenvalues are 1-fold.

For example, consider $A$ matrix as $$ A = \pmatrix{3 & 0 & 0\\ 0 & 5 & 0\\ 0 & 0 & 5}. $$ Then $A$ has one 1-fold eigenvalue 3 and one 2-fold eigenvalue 5, totally two distinct eigenvalues. But $A$ has three orthonormal eigenvectors with respect to eigenvalues 3, 5 and 5 $$ \pmatrix{1\\0\\0}, \pmatrix{0\\ 1 \\ 0}, \pmatrix{0 \\ 0 \\ 1}. $$