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 there is a matrix and a diagonal matrix such that . This happens if and only if has "n" eigenvectors which constitute a basis for . In this case, can be chosen to be the matrix with the "n" eigenvectors as columns, and thus a square root of is where is any square root of . Indeed, For example, the matrix formula_14 can be diagonalized as , where giving the square root When is symmetric, the diagonalizing matrix can be made an orthogonal matrix by suitably choosing the eigenvectors (see spectral theorem). Then the inverse of is simply the transpose,