“Square root of a 2 by 2 matrix”的意思、由来-开放百科全书

A square root of a 2×2 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 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]^[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 = AD – BC be its determinant. Let s be such that s² = δ, and t be such that t² = τ + 2s. That 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 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,

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 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

= 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,

, 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 matrix

where a = ±√A and d = ±√D; which, 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

has infinitely many symmetric rational square roots given by

where {{math|(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 zero

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.. 1980. “The Square Root of a 2 × 2 Matrix”. Mathematics Magazine 53 (4). Mathematical Association of America: 222–24. doi:10.2307/2689616.[https://www.jstor.org/stable/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):118–29
4. ^Mitchell, Douglas W. "Using Pythagorean triples to generate square roots of I₂". The Mathematical Gazette 87, November 2003, 499-500.