Let $V$ be a affine/projective variety on field $\mathbb{C}$ or $\mathbb{R}$. And let $M$ be a full-rank matrix from $\mathbb{C}^m\to\mathbb{C}^n$ or $\mathbb{R}^m\to\mathbb{R}^n$, respectively. Then I know that automorphism of projective space $\mathbb{CP}^m$ or $\mathbb{RP}^m$ is projective linear group.
But my question is :
Does $M$ preserve variety structure? If not, what are the conditions that a matrix can be a morphism between varieties?
I cannot find any references about this question, neither can I solve this question by myself, as I am a learning student.
Thanks for your help!