Square root of a 2 by 2 matrix

A square root of a 2×2 matrix M is another 2×2 matrix R such that M = R2, where R2 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 −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.

A general formula

The following is a general formula that applies to almost any 2 × 2 matrix.[1] Let the given matrix be

M = ( A B C D ) , {\displaystyle M={\begin{pmatrix}A&B\\C&D\end{pmatrix}},}
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. Let s be such that s2 = δ, and t be such that t2 = τ + 2s. That is,
s = ± δ , t = ± τ + 2 s . {\displaystyle s=\pm {\sqrt {\delta }},\qquad t=\pm {\sqrt {\tau +2s}}.}
Then, if t ≠ 0, a square root of M is
R = 1 t ( A + s B C D + s ) = 1 t ( M + s I ) . {\displaystyle R={\frac {1}{t}}{\begin{pmatrix}A+s&B\\C&D+s\end{pmatrix}}={\frac {1}{t}}\left(M+sI\right).}

Indeed, the square of R is

R 2 = 1 t 2 ( A 2 + B C + 2 s A + s 2 A B + B D + 2 s B C A + D C + 2 s C C B + D 2 + 2 s D + s 2 ) = 1 t 2 ( A 2 + B C + 2 s A + A D − B C A B + B D + 2 s B A C + C D + 2 s C B C + D 2 + 2 s D + A D − B C ) = 1 A + D + 2 s ( A ( A + D + 2 s ) B ( A + D + 2 s ) C ( A + D + 2 s ) D ( A + D + 2 s ) ) = M . {\displaystyle {\begin{aligned}R^{2}&={\frac {1}{t^{2}}}{\begin{pmatrix}A^{2}+BC+2sA+s^{2}&AB+BD+2sB\\CA+DC+2sC&CB+D^{2}+2sD+s^{2}\end{pmatrix}}\\[1ex]&={\frac {1}{t^{2}}}{\begin{pmatrix}A^{2}+BC+2sA+AD-BC&AB+BD+2sB\\AC+CD+2sC&BC+D^{2}+2sD+AD-BC\end{pmatrix}}\\[1ex]&={\frac {1}{A+D+2s}}{\begin{pmatrix}A(A+D+2s)&B(A+D+2s)\\C(A+D+2s)&D(A+D+2s)\end{pmatrix}}=M.\end{aligned}}}

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.

Special cases of the formula

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,

R = ± 1 t ( A B C D ) = ± 1 t M , {\displaystyle R=\pm {\frac {1}{t}}{\begin{pmatrix}A&B\\C&D\end{pmatrix}}=\pm {\frac {1}{t}}M,}
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. In that case, the two roots are

R = ± 1 t ( A + s B C D + s ) = ± 1 t ( M + s I ) , {\displaystyle R=\pm {\frac {1}{t}}{\begin{pmatrix}A+s&B\\C&D+s\end{pmatrix}}=\pm {\frac {1}{t}}\left(M+sI\right),}
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

R = ( 0 0 c 0 ) and R = ( 0 b 0 0 ) , {\displaystyle R={\begin{pmatrix}0&0\\c&0\end{pmatrix}}\quad {\text{and}}\quad R={\begin{pmatrix}0&b\\0&0\end{pmatrix}},}

where b and c are arbitrary real or complex values. Otherwise M has no square root.

Formulas for special matrices

Idempotent matrix

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.

Exponential matrix

If the matrix M can be expressed as real multiple of the exponent of some matrix A, M = r exp ⁡ ( A ) {\displaystyle M=r\exp(A)} , then two of its square roots are ± r exp ⁡ ( 1 2 A ) {\displaystyle \pm {\sqrt {r}}\exp \left({\tfrac {1}{2}}A\right)} . In this case the square root is real.[2]

Diagonal matrix

If M is diagonal (that is, B = C = 0), one can use the simplified formula

R = ( a 0 0 d ) , {\displaystyle R={\begin{pmatrix}a&0\\0&d\end{pmatrix}},}

where a = ±√A, and d = ±√D. 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.

Identity matrix

Because it has duplicate eigenvalues, the 2×2 identity matrix ( 1 0 0 1 ) {\displaystyle \left({\begin{smallmatrix}1&0\\0&1\end{smallmatrix}}\right)} has infinitely many symmetric rational square roots given by

1 t ( s r r − s )  and  ( ± 1 0 0 ± 1 ) , {\displaystyle {\frac {1}{t}}{\begin{pmatrix}s&r\\r&-s\end{pmatrix}}{\text{ and }}{\begin{pmatrix}\pm 1&0\\0&\pm 1\end{pmatrix}},}
where (r, s, t) are any complex numbers such that r 2 + s 2 = t 2 . {\displaystyle r^{2}+s^{2}=t^{2}.} [3]

Matrix with one off-diagonal zero

If B is zero, but A and D are not both zero, one can use

R = ( a 0 C a + d d ) . {\displaystyle R={\begin{pmatrix}a&0\\{\frac {C}{a+d}}&d\end{pmatrix}}.}

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.

References

  1. ^ Levinger, Bernard W. (September 1980), "The square root of a 2 × 2 {\displaystyle 2\times 2} matrix", Mathematics Magazine, 53 (4): 222–224, doi:10.1080/0025570X.1980.11976858, JSTOR 2689616
  2. ^ Harkin, Anthony A.; Harkin, Joseph B. (2004), "Geometry of generalized complex numbers" (PDF), Mathematics Magazine, 77 (2): 118–129, doi:10.1080/0025570X.2004.11953236, JSTOR 3219099, MR 1573734
  3. ^ Mitchell, Douglas W. (November 2003), "87.57 Using Pythagorean triples to generate square roots of I 2 {\displaystyle I_{2}} ", The Mathematical Gazette, 87 (510): 499–500, doi:10.1017/S0025557200173723, JSTOR 3621289