sqrtJJ1

sqrtJJ1 is a notational convention in linear algebra for the principal square root of a Hermitian positive-definite matrix referred to as JJ1. Specifically, sqrtJJ1 denotes the unique Hermitian positive-definite matrix S that satisfies S^2 = JJ1. When JJ1 is real symmetric positive-definite, sqrtJJ1 coincides with the standard real symmetric square root; for complex matrices it is defined via spectral decomposition or via the functional calculus.

Computational methods: The most common approach is to perform an eigen decomposition JJ1 = Q Λ Q*, with Λ a diagonal matrix of eigenvalues. Then sqrtJJ1 = Q sqrt(Λ) Q*. For large or sparse matrices, iterative methods such as Denman–Beavers or Newton–Schulz iterations, combined with scaling and squaring, are used. Factorizations like Cholesky can also be used indirectly when JJ1 is PD, but they do not generally produce the square root directly.

Properties: If JJ1 is Hermitian positive definite, sqrtJJ1 is Hermitian positive definite, with eigenvalues equal to the square roots of those of JJ1. The operation is continuous on the set of PD matrices, and is stable under small perturbations. If JJ1 has zero eigenvalues (positive semidefinite), a pseudoinverse-based or generalized square root is defined, but the standard principal square root is not unique in that case.

Applications: Matrix square roots arise in solving differential equations, covariance matrix transformations, diffusion processes, Kalman filtering, and control theory, as well as in numerical algorithms that require a stable square root of a PD matrix.

See also: matrix square root; eigen decomposition; Cholesky decomposition; Denman–Beavers iteration.