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 formulas that give up to four square roots, if the matrix has any roots. If is a diagonal "n" × "n" matrix, one can obtain a square root by taking a diagonal matrix , where each element along the diagonal is a square root of the corresponding element of . If the diagonal elements of "D" are real and non-negative, and the square roots are taken with non-negative sign, the matrix will be the principal root of . If a matrix is idempotent, one of its square roots is the matrix itself. An "n" × "n" matrix is diagonalizable if