Square root any nonnegative number, then the principal square root of −"x" is The right side (as well as its negative) is indeed a square root of −"x", since For every non-zero complex number "z" there exist precisely two numbers "w" such that : the principal square root of "z" (defined below), and its negative. There are two complex numbers that square to a given arbitrary non-zero imaginary number formula_11 with real formula_12 where the two-digit pattern {3, 6} repeats over and over again in the partial denominators. Since , the above is also identical to the following generalized continued fractions: The
Square root of a matrix is an "n" × "n" complex matrix "ρ" that is positive semidefinite with trace 1. If "ρ" can be expressed as where formula_36 and ∑ "p" = 1, the set is said to be an ensemble that describes the mixed state "ρ". Notice {"v"} is not required to be orthogonal. Different ensembles describing the state "ρ" are related by unitary operators, via the square roots of "ρ". For instance, suppose The trace 1 condition means Let and "v" be the normalized "a". We see that gives the mixed state "ρ". In the Unscented Kalman Filter (UKF), the square root of