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 to converge. In linear algebra and operator theory, given a bounded positive semidefinite operator (a non-negative operator) "T" on a complex Hilbert space, "B" is a square root of "T" if "T" = "B* B", where "B*" denotes the Hermitian adjoint of "B". According to the spectral theorem, the continuous functional calculus can be applied to obtain an operator "T" such that "T" is itself positive and ("T") = "T". The operator "T" is the unique non-negative square root of "T". A bounded non-negative operator on a complex Hilbert space is self adjoint by definition. So "T" = ("T")* "T".