Help me please,
I tried to prove, that JNF of matrix $ A + \alpha I$ is equal to matrix $ A_j + \alpha I$ where $A_j$ is JNF of A.
Is it true, that $ A_j + \alpha I$ - JNF by definition? Because, it is block diagonal matrix and every block are Jordan blocks.
If $S^{-1}AS=B$ then $S^{-1}(A+\alpha I)S=S^{-1}AS+S^{-1}\alpha I S=B+\alpha I$.
If $B$ is the Jordan canonical form of $A$, then by inspection $B+\alpha I$ is in Jordan canonical form, and hence is the Jordan canonical form of $A+\alpha I$.