Is it true that for any $A\in M(n,\mathbb{C})$there exist a $B\in M(n,\mathbb{C})$ such that $A = B^2$? I think this is not true (but I don't know nay example), and then is it possible to characterize such $A$?
$A = B^2$ for which matrix $A$?
10
$\begingroup$
linear-algebra
matrices
radicals
-
7See Wikipedia - [Square root of a matrix](http://en.wikipedia.org/wiki/Square_root_of_a_matrix). The case $n=2$ is discussed here: [For every matrix $A\in M_{2}( \mathbb{C}) $ there's $X\in M_{2}( \mathbb{C})$ such that $X^2=A$?](http://math.stackexchange.com/questions/57292/for-every-matrix-a-in-m-2-mathbbc-theres-x-in-m-2-mathbbc-s) – 2012-10-04
-
0I'll copy here also J.M.'s comment an older question [How to find the square root of a matrix?](http://math.stackexchange.com/questions/180638/how-to-find-the-square-root-of-a-matrix), which is now deleted. Higham's book *Functions of Matrices: Theory and Computation* devotes [an entire chapter](http://books.google.com/books?hl=en&id=S6gpNn1JmbgC&pg=PA133) to the subject. Which algorithm to use depends on the structure of your matrix, about which you have not yet said anything. – 2012-10-04
-
0by Starting to compute the Jourdan reduction of your matrix. you juste have to determine whether or not a matrix of the form $\lambda * I_n +N$ with $N$ a reduced nilpotent matrix has a square root or not. it seem that when $\lambda$ is non zero it's always the case. So in the end you have to find a way to know when a reduced nilpotent matrix has a square root or not, this seem to be a purely combinatorics question, and the answer will only depend on the dimension. (so the general answer will depend on the dimmension of the jordan block of the 0 eigenvalue) – 2012-10-04
-
0Thanks for all the comments. I still don't know sufficient condition but my understanding is much better now. – 2012-10-04