A square root of a 2×2 matrix is another 2×2 matrix such that , where stands for the matrix product of with itself. In general, there can be zero, two, four, or even an infinitude of square-root matrices. In many cases, such a matrix can be obtained by an explicit formula.
Square roots that are not the all-zeros matrix come in pairs: if is a square root of , then is also a square root of , since .
A 2×2 matrix with two distinct nonzero eigenvalues has four square roots. A positive-definite matrix has precisely one positive-definite square root.
A general formula The following is a general formula that applies to almost any 2 × 2 matrix. Let the given matrix be
where , , , and may be real or complex numbers. Furthermore, let be the trace of , and be its determinant. Let be such that , and be such that . That is,
Then, if , a square root of is
Indeed, the square of is
Note that may have complex entries even if is a real matrix; this will be the case, in particular, if the determinant is negative.
The general case of this formula is when is nonzero, and , in which case is nonzero, and is nonzero for each choice of sign of . Then the formula above will provide four distinct square roots , one for each choice of signs for and .
Special cases of the formula If the determinant is zero, but the trace is nonzero, the general formula above will give only two distinct solutions, corresponding to the two signs of . Namely,
where is any square root of the trace .
The formula also gives only two distinct solutions if is nonzero, and (the case of duplicate eigenvalues), in which case one of the choices for will make the denominator be zero. In that case, the two roots are
where is the square root of that makes nonzero, and is any square root of .
The formula above fails completely if and are both zero; that is, if , and , so that both the trace and the determinant of the matrix are zero. In this case, if is the null matrix (with ), then the null matrix is also a square root of , as is any matrix
where and are arbitrary real or complex values. Otherwise has no square root.
Formulas for special matrices
Idempotent matrix If is an idempotent matrix, meaning that , then if it is not the identity matrix, its determinant is zero, and its trace equals its rank, which (excluding the zero matrix) is . Then the above formula has and , giving and as two square roots of .
Exponential matrix If the matrix can be expressed as real multiple of the exponent of some matrix , , then two of its square roots are . In this case the square root is real.
Diagonal matrix If is diagonal (that is, ), one can use the simplified formula
where , and . This, for the various sign choices, gives four, two, or one distinct matrices, if none of, only one of, or both and are zero, respectively.
Identity matrix Because it has duplicate eigenvalues, the 2×2 identity matrix has infinitely many symmetric rational square roots given by
where are any complex numbers such that
Matrix with one off-diagonal zero If is zero, but and are not both zero, one can use
This formula will provide two solutions if or or , and four otherwise. A similar formula can be used when is zero, but and are not both zero. Real matrices with real square roots The algebra of 2×2 real matrices has three types of planar subalgebras. Each subalgebra admits the exponential map. If , then are square roots of . 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 since each quadrant contains one.
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