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 "UB". Indeed, take "B" = "T" to be the unique non-negative square root of "T". If "T" is strictly positive, then "B" is invertible, and so is unitary: If "T" is non-negative without being strictly positive, then the inverse of "B" cannot be defined, but the Moore–Penrose pseudoinverse "B" can be. In that case, the operator is a partial isometry, that is, a unitary operator from the range of "T" to itself. This can then be extended to a unitary operator "U" on the whole space by setting it equal to the identity on the kernel of "T". More generally,