Is there another name for matrices that have positive definite square root?
Motivation: some linear estimation algorithms for $Ax=b$, like Richardson's iteration, only work when $A$ is positive definite. Because these algorithms correspond to minimizing a quadratic form $B=A^TA$, this means that quadratic forms of this type are easier to minimize than general, wondering if there's another name for them.
The matrices that have positive-definite square roots are exactly the positive definite matrices. See e.g. Square root of Positive Definite Matrix (showing that any positive definite matrix has a positive definite square root), If $A$ is positive definite then so is$A^2$. (showing that the square of a positive definite matrix is positive definite).
The details of the proofs depend on the definitions, of course, but the "harder" direction of this equivalence (which, given basic theory as developed in most textbooks, is not all that hard), is usually the fact that a positive definite matrix necessarily has a positive definite square root. This fact is not an immediate consequence of most sets of definitions, but is instead often proved using some version of what people sometimes call the "spectral theorem" (roughly speaking: some characterization of the consequences of positive definiteness in terms of eigenspaces and eigenvalues, or an equivalent matrix factorization).
See also https://en.wikipedia.org/wiki/Definite_matrix (stating that "A Hermitian matrix $M$ is positive semidefinite if and only if there is a positive semidefinite matrix $B$ . . . satisfying $M = BB$," and linking to https://en.wikipedia.org/wiki/Square_root_of_a_matrix, which contains further discussion at varying levels of generality).
Another name may refer to self-adjoint positive linear operators. If you have a self-adjoint positive linear operator $\varphi>0$ you can take a square root of it: $\psi=\sqrt{\varphi}$. More information is in 'Polar decomposition' topic.
