This question seems to be very basic, but I couldn't find the answer on SE.
I want to know if a diagonalizable matrix $A$, i.e. $A = SDS^{-1}$, is also an injective matrix, i.e. $Av = 0$ if and only if $v = 0$ or also $\ker A = \{ 0 \}$.
I believe it can be simply proven by
- "a composition of injective functions is also injective" and
- "$S, S^{-1}$ are injective because they are invertible and $D$ is injective because it is diagonal",
but I want to know if I am correct.
I'm not sure about the injectiveness of $D$. If this is the dealbreaker, what are criteria for the injectiveness of $D$ proceeding from the properties of $A$?
I am asking because this would imply, that the only fixed point of a dynamical system
$ \dot{\vec{x}} = A \vec{x}$ with a diagonalizable matrix $A$ would be only $\vec x = 0$.
Best Answer
No, (the linear transformations given by) diagonal matrices are not injective in general, since a diagonal matrix can have zero entries on the diagonal (and a diagonalizable matrix can have $0$ as an eigenvalue). In fact, being diagonal or diagonalizable is irrelevant when it comes to being injective.
A square matrix describes an injective map exactly when $0$ is not an eigenvalue, or, equivalently, exactly when the determinant is nonzero. I don't think there is any simpler criterion.