Thread: The kth root of a zero matrix

1. The kth root of a zero matrix

I've never really studied roots of matrices before, but I am finding myself wanting to take the kth root of the nxn zero matrix in a maths problem of mine.

I understand that roots of matrices are not always unique, I was wondering what can be said of the kth root for the zero matrix? Is it the zero matrix itself? Are there several possible (maybe an infinite number) of roots?

My intuition says that it is a diagonal matrix, so I can just take the kth root of everything on the leading diagonal. Then this will be the zero matrix. But I also know that such an attempt on the identity matrix does not yield a unique root for k=2. For example, you can use any Pythagorean triple and plug it in to a certain formulaic matrix and get a completely unexpected looking root of the identity, so I was wondering what happens for zero?

Thanks.

2. Re: The kth root of a zero matrix

Yes, matrices can have many "k"th roots just as a number can have many "k"th roots. However, "0" is a special case. Just as the only kth root of 0, for any k, is 0 itself, so the 0 matrix is the only root of the 0 matrix.