TITLE: What do zero eigenvalues mean?
QUESTION [35 upvotes]: What is the geometric meaning of a 3x3 matrix having all three eigenvalues as zero? I have interpretations in mind for 0, 1, and 2 eigenvalues being zero, but what about all of them?

REPLY [3 votes]: If 0 is an eigenvalue, then the nullspace is non-trivial and the matrix is not invertible. Therefore all the equivalent statements given by the invertible matrix theorem that apply to only invertible matrices are false.