Let be a field whose characteristic is not 2. Characterize those matrices over which have square roots. That is, characterize such that for some matrix .
We claim that has a square root if and only if the following hold.
- Every eigenvalue of is square in .
- The Jordan blocks of having eigenvalue 0 can be paired up in such a way that the sizes of the blocks in each pair differ by 0 or 1.
First we tackle the ‘if’ part. Suppose is in Jordan canonical form, where are the Jordan blocks having nonzero eigenvalue and and are the blocks with eigenvalue 0 such that the sizes of and differ by 0 or 1. As we showed in this previous exercise, is the Jordan canonical form of the square of the Jordan block with eigenvalue , and is the Jordan canonical form of the square of the Jordan block whose size is with eigenvalue 0. So we have , and so is a square.
Conversely, suppose is a square, and say is in Jordan canonical form. Now . Now say is in Jordan canonical form; then . By the previous exercise, the nonzero eigenvalues of are square in , and the Jordan blocks having eigenvalue 0 can be paired so that the sizes in each pair differ by 0 or 1.