Supriya Ghosh (Editor)

Square root of a 2 by 2 matrix

Updated on
Edit
Like
Comment
Share on FacebookTweet on TwitterShare on LinkedInShare on Reddit

A square root of a 2 by 2 matrix M is another 2 by 2 matrix R such that M = R2, where R2 stands for the matrix product of R with itself. In general there can be no, two, four or even an infinitude of square root matrices. In many cases such a matrix R can be obtained by an explicit formula.

Contents

A 2 × 2 matrix with two distinct nonzero eigenvalues has four square roots. A positive-definite matrix has precisely one positive-definite square root.

Square roots of a matrix of any dimension 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.

One formula

Let

M = ( A B C D )

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 .

Then, if t ≠ 0, a square root of M is

R = 1 t ( A + s B C D + s ) .

Indeed, the square of R is

R 2 = 1 t 2 ( ( A + s ) 2 + B C ( A + s ) B + B ( D + s ) C ( A + s ) + ( D + s ) C ( D + s ) 2 + B C ) = 1 A + D + 2 s ( A ( A + D + 2 s ) ( A + D + 2 s ) B C ( A + D + 2 s ) D ( A + D + 2 s ) ) = M .

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. Also, note that R is positive when s>0 and t>0.

Special cases of the formula

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.

In general, 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 formula will give only two distinct solutions. It 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.

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 are

R = ( 0 0 c 0 ) and R = ( 0 b 0 0 )

for any real or complex values of b and c. Otherwise M has no square root.

Diagonal matrix

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

R = ( a 0 0 d )

where a = ±√A and d = ±√D; which, depending on the 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 ) has infinitely many symmetric rational square roots given by

1 t ( s r r − s ) , 1 t ( s − r − r − s ) , 1 t ( − s r r s ) , 1 t ( − s − r − r s ) , ( 1 0 0 ± 1 ) , and ( − 1 0 0 ± 1 ) ,

where (r, s, t) is any Pythagorean triple—that is, any set of positive integers such that r 2 + s 2 = t 2 . In addition, any non-integer, irrational, or complex values of r, s, t satisfying r 2 + s 2 = t 2 give square root matrices. The identity matrix also has infinitely many non-symmetric square roots.

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 ) .

This formula will provide two solutions if A = D, and four otherwise. A similar formula can be used when C is zero but A and D are not both zero.

References

Square root of a 2 by 2 matrix Wikipedia
(Text) CC BY-SA