if y is a left eigenvector of A, then y*A = λy*. therefore, (y*A)* = (λy*)*, that is:

A*y = λ*y.

(writing λ* for the complex conjugate of λ).

the trick is noticing that if:

then:

so:

for the second question, note that y*A = λy*, means that y*(A - λI) = 0^t (the 0 row-vector), so

that is: so that: