# Do all square matrices have eigenvectors?

I came across a video lecture in which the professor stated that there may or may not be any eigenvectors for a given linear transformation.

But I had previously thought every square matrix has eigenvectors.

has no eigenvalues at all (i.e., over $\;\Bbb R\;$ ), yet the very same matrix defined over the complex field $\;\Bbb C\;$ has two eigenvalues: $\;\pm i\;$