When does a real matrix have a real square root?

Question

Given a real symmetric positive-definite matrix $A$, it is easy to show that there is a real square root: the spectral theorem says that there will be a real eigenbasis, and that reduces the problem to taking the square root of the diagonalization. In fact, we don't need symmetry, assuming we have some other way of determining that our matrix is diagonalizable with real and positive eigenvalues.

However, there is no need for the eigenvalues to be real and positive. For example, by using the Taylor series for $\sqrt{1+x}$, one can show that a real $2\times 2$ matrix $A$ will have a real square root if $\operatorname{tr}(A)>0$ and $0<\det(A)<\operatorname{tr}(A)^2/2$.

Are there good general necessary or sufficient (or both) conditions under which a real square matrix has a real square root?

Answer 1

1. Let $A\in M_n(\mathbb{C})$; we consider the non-increasing sequence: $d_i=dim(\ker(A^i))-dim(\ker(A^{i-1}))$.

$A$ is a square IFF $A$ satisfies the following condition (P) about its iterated kernels (if $A$ is invertible, then there are no conditions!)

(P) (cf. Cross, Lancaster, Square roots of complex matrices, Linear and Multilinear Algebra): $(d_i)_i$ does not contain two successive occurrences of the same odd integer.

Note that, in Topics in Matrix Analysis, p. 472, Horn and Johnson add a condition which is useless.

2. Let $A\in M_n(\mathbb{R})$. $A$ is a square over $\mathbb{R}$ IFF $A$ satisfies (P) and the following condition (Q) concerning the $<0$ eigenvalues of $A$.

(Q) $A$ has an even number of Jordan blocks of each size for every $<0$ eigenvalue. (Use the real Schur decomposition. cf. Functions of matrices, Higham, p.17 or Horn and Johnson -above-).

EDIT. Answer to Aaron. 1. The supplementary condition by H&J, can be rewritten: "if $dim(\ker(A))$ is odd, then $dim(\ker(A^2))<2dim(\ker(A))$". Note that $dim(\ker(A^2))\leq 2dim(\ker(A))$ is always true. Then it suffices to assume that $dim(\ker(A^2))=2dim(\ker(A))$; then $d_1=d_2$ are odd and, according to the other conditions, the square root of $A$ does not exist.

2. The problem only stands for real $<0$ eigenvalues; indeed, if $\lambda$ is an eigenvalue of $A$ then $\bar{\lambda}$ too, and the dimensions of the Jordan blocks associated to $\lambda$ are the same as those of $\bar{\lambda}$; it is not difficult to find a square root of $diag(\lambda I+J_k,\bar{\lambda}I+J_k)$ when $\lambda\notin \mathbb{R}$ or of $\lambda I+J_k$ when $\lambda >0$ or of $diag(\lambda I+J_k,\lambda I+J_k)$ when $\lambda<0$.