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 finitely many terms will be non-zero and It suffices to check uniqueness for a Jordan block with λ = 1. The square constructed above has the form "S" = "I" + "L" where "L" is polynomial in "N" without constant term. Any other square root "T" with positive eigenvalues has the form "T" = "I" + "M" with nilpotent, commuting with "N" and hence "L". But then . Since "L" and commute, the matrix is nilpotent and is invertible with inverse given by a Neumann series. Hence "L" = . If is a matrix with positive eigenvalues and minimal polynomial