[RESOLVED] Is a zero matrix nilpotent?

[IliaPreston]

I have checked a number of books and a number of Internet websites for nilpotent matrices.
For example this one:
https://en.wikipedia.org/wiki/Nilpotent_matrix

But, none of them make any mention of a zero matrix itself being nilpotent or not.

In all places that I have checked, they say if N is a matrix and k is a positive integer and N^k = 0 then N is a nilpotent matrix.

According to the above definition, the zero matrix itself should be nilpotent, that is when k = 1.
Obviously k = 1 is a positive integer.

But, to me it doesn't make sense.
I guess the above definition misses the condition k > 1 by mistake and that it should say that if N <> 0 and for an integer k > 1, if N^k = 0 then N is nilpotent.
The rational for my guess is that nilpotence is a special characteristic of a matrix, and therefore deserves to be called just that (nilpotent), but if k = 1 then the matrix itself is zero and therfore nothing is special about it to deserve that kind of categorization (as nilpotent)

So, what do you think?
A zero matrix itself is nilpotent or not?
Can k be equal 1 and the matrix still be called nilpotent or not?

Please advise.
Thanks.

[wqweto]

https://chatgpt.com/share/67ee7e96-6...f-ad826803d00e

> According to the above definition, the zero matrix itself should be nilpotent, that is when k = 1.

No, not only. This is true for *any* k.

cheers,
</wqw>

[OptionBase1]

I have checked a number of books and a number of Internet websites for nilpotent matrices.
For example this one:
https://en.wikipedia.org/wiki/Nilpotent_matrix

But, none of them make any mention of a zero matrix itself being nilpotent or not.

Really? From the wikipedia article you linked to:

The only nilpotent diagonalizable matrix is the zero matrix.

Pretty clear from that that the zero matrix is considered nilpotent. I assume that the terminology traditionally used for it would be to say that it is trivially nilpotent.

[IliaPreston]

Really? From the wikipedia article you linked to:



Pretty clear from that that the zero matrix is considered nilpotent. I assume that the terminology traditionally used for it would be to say that it is trivially nilpotent.

Wow! I don't know why I initially missed it.
Thanks a lot for pointing that out.

[IliaPreston]

https://chatgpt.com/share/67ee7e96-6...f-ad826803d00e

> According to the above definition, the zero matrix itself should be nilpotent, that is when k = 1.

No, not only. This is true for *any* k.

cheers,
</wqw>

Thanks for the help.