For any scalar $\alpha$ and square matrix $P$, the following are equivalent:


$\alpha P$ has an eigenvalue $1$
$\alpha Pv = v$ for some non-zero vector $v$
$(I - \alpha P)v = 0$ for some non-zero vector $v$
$I - \alpha P$ is not invertible


Eigenvalues of a transition probability matrix
What kind of non-technical training course do you suggest for a programmer? Example could be public speaking course, presentation skill, English, business writing, or anything not related to programming or software engineering itself.
