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 Conversely, it is trivially true that every operator of the form "B* B" is non-negative. Therefore, an operator "T" is non-negative if and only if "T" = "B* B" for some "B" (equivalently, "T" = "CC*" for some "C"). The Cholesky factorization provides another particular example of square root, which should not be confused with the unique non-negative square root. If "T" is a non-negative operator on a finite-dimensional Hilbert space, then all square roots of "T" are related by unitary transformations. More precisely, if "T" = "A*A" = "B*B", then there exists a unitary "U" such that "A" =