Adjoint vectors of a matrix

Question

Consider a real square matrix $A$ (a monodromy matrix of a Hamiltonian system, but it doesn't matter here).

I found on the web two meanings for the adjoint vectors of such matrix:

1. If $Y$ is an eigenvector of $A$, and $AY=e^{i\phi}Y$, then an adjoint vector of $A$ is $AZ=e^{i\phi}Z+Y$
2. $(A-\lambda I)Y=Z$

Are those two definitions compatible?

Answer 1

There is very likely a typo in your number 2: if $AY=\lambda Y$, then $(A-\lambda I)Y=0$, so your second equality is $Z=0$.

If exchange the roles, on the other hand, you get $(A-\lambda I)Z=Y$, which exactly the same equality as in 1.