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 formulaEdit

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

 

where A, B, C, and D may be real or complex numbers. Furthermore, let τ = A + D be the trace of M, and δ = ADBC be its determinant. Let s be such that s2 = δ, and t be such that t2 = τ + 2s. That is,

 

Then, if t ≠ 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 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 formulaEdit

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. In that case, the two roots are

 

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.

Formulas for special matricesEdit

Idempotent matrixEdit

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 matrixEdit

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 and can be interpreted as the square root of a type of complex number.[3]

Diagonal matrixEdit

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

 

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 matrixEdit

Because it has duplicate eigenvalues, the 2×2 identity matrix   has infinitely many symmetric rational square roots given by

 

where (r, s, t) is any Pythagorean triple—that is, any set of positive integers such that  [4] In addition, any non-integer, irrational, or complex values of r, s, t satisfying   give square-root matrices. The identity matrix also has infinitely many non-symmetric square roots.

Matrix with one off-diagonal zeroEdit

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.

ReferencesEdit

  1. ^ Levinger, Bernard W.. 1980. “The Square Root of a 2 × 2 Matrix”. Mathematics Magazine 53 (4). Mathematical Association of America: 222–224. doi:10.2307/2689616.
  2. ^ P. C. Somayya (1997), Root of a 2x2 Matrix, The Mathematics Education, Vol. XXXI, no. 1. Siwan, Bihar State. INDIA.
  3. ^ Anthony A. Harkin & Joseph B. Harkin (2004) Geometry of Generalized Complex Numbers, Mathematics Magazine 77(2):118–129.
  4. ^ Mitchell, Douglas W. "Using Pythagorean triples to generate square roots of I2". The Mathematical Gazette 87, November 2003, 499–500.