sqrtTM

sqrtTM is not a standard, universally defined term in mathematics or science. In many texts, TM is simply a label for a quantity, a matrix, or a parameter, and sqrtTM would denote the square root of that quantity as interpreted in the given context. Because there is no single canonical meaning, the precise interpretation depends on what TM represents in a particular discussion.

For scalar TM, the square root is the nonnegative principal root when TM is a nonnegative real number. If TM is negative or complex, the square root is defined in the corresponding complex sense, with a chosen branch if a complex plane cut is specified. In practice, TM as a real nonnegative parameter yields sqrtTM as a real, nonnegative value; for other values, sqrtTM becomes complex.

For matrices or linear operators, sqrtTM denotes a matrix B such that B squared equals TM, i.e., B^2 = TM. Existence and uniqueness depend on the properties of TM. If TM is Hermitian positive semidefinite, there exists a unique Hermitian positive semidefinite square root. In general, a complex or non-Hermitian matrix may have zero, one, or multiple square roots, and some matrices may have none. Computational methods include eigenvalue decomposition, where TM = VDV^{-1} and sqrtTM = V sqrt(D) V^{-1}, Schur decomposition-based algorithms, or iterative schemes like the Denman–Beavers iteration. Special care is needed when TM has repeated eigenvalues or is defective.

Beyond finite-dimensional matrices, the concept extends to operators on function spaces and to other algebraic structures that admit a meaningful square root operation under spectral or functional calculus. When encountering sqrtTM, the key is to identify what TM represents and apply the corresponding scalar or operator square-root rules, noting domain, branch choices, and existence conditions for the object in question.