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 −R is also a square root of M, since (−R)(−R) = (−1)(−1)(RR) = R2 = M.A 2×2 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 × 2 matrix.
where A, B, C, and D may be real or complex numbers.
Furthermore, let τ = A + D be the trace of M, and δ = AD − BC be its determinant.
Then, if t ≠ 0, a square root of M 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 δ is negative.
The general case of this formula is when δ is nonzero, and τ2 ≠ 4δ, 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 δ is zero, but the trace τ is 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 τ.
The formula also gives only two distinct solutions if δ is nonzero, and τ2 = 4δ (the case of duplicate eigenvalues), in which case one of the choices for s will make the denominator t be zero.
where s is the square root of δ that makes τ − 2s nonzero, and t is any square root of τ − 2s.
The formula above fails completely if δ and τ are both zero; that is, if D = −A, and A2 = −BC, 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 τ = 1, giving M and −M 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.
[2] If M is diagonal (that is, B = C = 0), one can use the simplified formula
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 2×2 identity matrix
has infinitely many symmetric rational square roots given by
This formula will provide two solutions if A = D or A = 0 or D = 0, and four otherwise.
The algebra M(2, R) of 2x2 real matrices has three types of planar subalgebras.
Each subalgebra admits the exponential map.
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.