[Math] Nilpotent Matrix is Similar to a block diagonal matrix

Question

Prove that any nilpotent matrix is similar to a block diagonal matrix whose blocks are matrices with 1's along the first super diagonal and 0's elsewhere.

I'm not sure where to start exactly. Any guidance would be helpful!

Answer 1

My answer is based on Wikipedia articles "Nilpotent matrix" and "Jordan normal form."

Every square matrix A over an algebraically-closed field has a Jordan canonical form J, which has zeros everywhere except A's eigenvalues as diagonal entries and possibly ones on the super diagonal. A is similar to J because there exists an invertible matrix P such that A = P J P ⁻¹. Because zero is the only eigenvalue of a nilpotent matrix, J has the form you want.