Zero Matrix Nilpotency: Defined & Explained

[*best&sweetest*]

Can a zero matrix be considered as nilpotent matrix?
Zero matrix raised to any positive power is zero matrix, so can it be considered nilpotent (with index of nilpotency being 1)? I have read the definition of the nilpotent matrix and it doesn't that say that a matrix has to different from 0, but I'm still confused...

 

[dimachka]

sure, you can say that the zero matrix is nilpotent, but that would be considered the trivial case. i.e., if you are asked to find a nilpotent matrix satisfying some properties, using the zero matrix will probably not get you credit for solving the question.

 

[HallsofIvy]

Why are you confused? Another characterization of "nilpotent" is that a matrix is nilpotent if and only if its eigenvalues are all 0. That is certainly true of the 0 matrix. The 0 matrix is definitely nilpotent.

 

[*best&sweetest*]

Thank you!