A square root of a 2ÃÂ2 matrix M is another 2ÃÂ2 matrix R such that M = R<sup>2</sup>, where R<sup>2</sup> stands for the matrix product of R with itself. In general, there can be zero, two, four, or even an infinitude of square-root matrices. In many cases, such a matrix R can be obtained by an explicit formula.
Square roots that are not the all-zeros matrix come in pairs: if R is a square root of M, then âÂÂR is also a square root of M, since (âÂÂR)(âÂÂR) = (âÂÂ1)(âÂÂ1)(RR) = R<sup>2</sup> = M.<br>A 2ÃÂ2 matrix with two distinct nonzero eigenvalues has four square roots. A positive-definite matrix has precisely one positive-definite square root.
The following is a general formula that applies to almost any 2 ÃÂ 2 matrix. Let the given matrix be
where A, B, C, and D may be real or complex numbers. Furthermore, let à= A + D be the trace of M, and ô = AD â BC be its determinant. Let s be such that s<sup>2</sup> = ô, and t be such that t<sup>2</sup> = à+ 2s. That is,
Then, if t â 0, a square root of M is
Indeed, the square of R is
Note that R may have complex entries even if M is a real matrix; this will be the case, in particular, if the determinant ô is negative.
The general case of this formula is when ô is nonzero, and ÃÂ<sup>2</sup> â 4ô, in which case s is nonzero, and t is nonzero for each choice of sign of s. Then the formula above will provide four distinct square roots R, one for each choice of signs for s and t.
If the determinant ô is zero, but the trace àis nonzero, the general formula above will give only two distinct solutions, corresponding to the two signs of t. Namely,
where t is any square root of the trace ÃÂ.
The formula also gives only two distinct solutions if ô is nonzero, and ÃÂ<sup>2</sup> = 4ô (the case of duplicate eigenvalues), in which case one of the choices for s will make the denominator t be zero. In that case, the two roots are
where s is the square root of ô that makes ÃÂàâÂÂà2s nonzero, and t is any square root of ÃÂàâÂÂà2s.
The formula above fails completely if ô and àare both zero; that is, if D = âÂÂA, and A<sup>2</sup> = âÂÂBC, so that both the trace and the determinant of the matrix are zero. In this case, if M is the null matrix (with A = B = C = D = 0), then the null matrix is also a square root of M, as is any matrix
where b and c are arbitrary real or complex values. Otherwise M has no square root.
If M is an idempotent matrix, meaning that MM = M, then if it is not the identity matrix, its determinant is zero, and its trace equals its rank, which (excluding the zero matrix) is 1. Then the above formula has s = 0 and à= 1, giving M and âÂÂM as two square roots of M.
If the matrix M can be expressed as real multiple of the exponent of some matrix A, , then two of its square roots are . In this case the square root is real.
If M is diagonal (that is, B = C = 0), one can use the simplified formula
where a = ñâÂÂA, and d = ñâÂÂD. This, for the various sign choices, gives four, two, or one distinct matrices, if none of, only one of, or both A and D are zero, respectively.
Because it has duplicate eigenvalues, the 2ÃÂ2 identity matrix has infinitely many symmetric rational square roots given by
where are any complex numbers such that
If B is zero, but A and D are not both zero, one can use
This formula will provide two solutions if A = D or A = 0 or D = 0, and four otherwise. A similar formula can be used when C is zero, but A and D are not both zero.
The algebra M(2, R) of 2x2 real matrices has three types of planar subalgebras. Each subalgebra admits the exponential map. If are square roots of p. The condition that the matrix is the image under exp limits it to half the plane of dual numbers, and to a quarter of the plane of split complex numbers, but does not constrain ordinary complex planes since the exponential mapping covers them. In the split-complex case there are two more square roots of p since each quadrant contains one.