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Cramer rule for quaternionic linear equations in quaternionic quantum theory
Vol. 57 (2006)
REPORTS ON MATHEMATICAL PHYSICS
No. 3
CRAMER RULE FOR QUATERNIONIC LINEAR EQUATIONS IN QUATERNIONIC QUANTUM THEORY* TONGSONG JIANG Department of Mathematics, Linyi Normal University, Shandong 276005, China Department of Computer Science and Technology, Shandong University, Jinan 250100, China (e-mail: [email protected]) (Received September 28, 2005)
By means of a complex representation of a quaternion matrix and a companion vector, this paper introduces a new definition of determinant for a quaternion matrix, derives a technique of finding an inverse matrix of a quaternion invertible matrix, and gives a Cramer rule for quaternionic linear equations in quaternionic quantum theory.
Keywords: Cromer rule, complex representation, companion vector, quatemionic linear equation.
1. Introduction In the study of quaternionic quantum mechanics and some other applications of quaternions [1-6], one often encounters the problem of solutions of quaternionic linear equations. Because of noncommutativity of quaternions, the solutions of quaternionic linear equations are more difficult. In papers [7, 8], by means of a complex representation and a companion vector, we have studied the problems of solutions of quatemionic linear equations and algorithms for eigenvalues and eigenvectors of a quaternion matrix, respectively, in quaternionic quantum theory. The paper [7] gave not only the necessary and sufficient conditions for the existence of solutions of quaternionic linear equations, but also a technique of finding solutions to quaternionic linear equations. This paper, by means of a complex representation and a companion vector, introduces a new definition of determinant for a quaternion matrix, gives an algorithm for inverse matrix of a quaternion invertible matrix, and gives Cramer rule for quatemionic linear equations in the quaternionic quantum theory. Let R denote the field of real numbers, C = {a + bvCL-fla, b ~ R} the field of complex numbers, and Q the field of quaternions. For x ~ C, ~ is the conjugate of x. Let IFm×n denote the set of m x n matrices on a field Y. For any A ~ C nxn, A r, det(A) and adj(A) denote the transpose, the determinant and the adjoint of the matrix A, respectively. *Supported by the National Natural Science Foundation of China and Shandong Natural Science Foundation of China (Y2005A12). [463]
464
T. JIANG
We first recall the complex representation and the companion vector [7, 8]. For any quaternion x = Xo + xli + x2j + x3k = y + z j 6 Q, in which xi 6 1R, and i 2 = j2 = k 2 = - 1 , ij = - j i = k, and a quaternion matrix A E Qm×n, the complex representations of x and A were defined respectively by
x2+x3 l[yz ]
xl = E xO+Xl
=
--X2 -I- X3"X,/'~
XO - - X l % / ~
--~
~
c2x2
(1.1)
y
and Af = (af) = [[
Yst I_1_ - g , ,
Zst ] ] ~ Y,, 13
Qf __c2m×2 n. c
(1.2)
If c~ = (xl, x2 . . . . . X2n) T ~ C 2n×1, then the companion vector otc of a vector ot was defined as oec = (-x2, xl, - x 4 , x3 . . . . . -Xzn, ~2n-1) r E C 2n×1. For any quaternion matrix A c Qm×n, by the definition of complex representation of a quaternion matrix there exist complex vectors oq, or2. . . . . oen such that A f = c c . . . , otn, otc). (Ofl,Oll,Ot2,0/2,
From [7, 8] we know that if A ~ Qm×~ and B E Qn×S, then (1.3)
(AB) f = Af B f,
and if A E Q~×n, then A is nonsingular if and and (Af) -1 = (A-l) f. Let A 6 Qn×n, then the real representation A f appear in pairs, and the imaginary conjugate pairs, respectively. The following result follows immediately from the
only if A f is nonsingular, eigenvalues of the complex eigenvalues of A f appear in statement above.
PROPOSITION 1.1. Let A ~ Qnxn and A f be its complex representation. Then the determinant det(A f) o f A f is a nonnegative real number.
2.
Determinant of quaternion matrices
In this section we define a determinant of a quatemion matrix by means of its complex representation and companion vector. DEFINITION 2.1. Let A ~ Qn×n and A f be its complex representation. Then determinant det(A) of the quaternion matrix A is defined as det(A) = det(Af).
(2.1)
From this definition and Proposition 1.1 we easily know that the determinant of a quaternion matrix is a nonnegative real number, and if A c Qn×n, B ~ Q~×n, then det(AB) = det(A) det(B).
(2.2)
Let A c Qn×n. Then it is easy to show by a direct calculation that adj(A f) E Q f . Now we introduce the concept of adjoint matrix of a quaternion matrix A by means of complex representation.
CRAMER RULE FOR QUATERNIONIC LINEAR EQUATIONS... D E F I N I T I O N 2.2. Let A c Qn×, and adjoint adj(A) of A is defined as
Af
be its complex representation. Then the
adj(A) = (adj(Af )) f-~. For any A
~
465
(2.3)
Qnxn we have (2.4)
(A adj(A)) / = A f (adj(A)) f = det(Af )I2,,
and by the definitions of complex representation and determinant of a quaternion matrix, A adj(a) = det(af)I n = det(A)In. Similarly, adj(A)A = det(A)In. PROPOSITION 2.1. Let A ~ (~×~, then (1) A adj(a) = a d j ( a ) a = det(a)In; (2) A is an invertible quaternion matrix if and only if det(A) # 0, in which case 1 A -1 -- - adj(A). (2.5) det(A) Proposition 2.1 not only gives the necessary and sufficient conditions for a quaternion matrix to be invertible, but also gives a technique of finding the inverse by means of the complex representation of the matrix.
3.
Cramer rule for quaternionic linear equations In this section we give Cramer rule for quaternionic linear equations in quaternionic quantum mechanics and quantum fields by means of complex representation of the quaternion matrix. If A 6 Qn×n, 13 6 (~n×l, then by the definition of complex representation, Ax = 13 if and only if A f x f = 13f. That is, Ax = 13 has a solution x if and only if A f Y = 13f has a solution Y = x f. If a quaternion matrix A is invertible, i.e. det(A) = det(A f) # 0, then the complex linear equations A f Y = 13f have a unique solution x f. Since adj(A f) E Qf, so we may put adj(Af)=[[
_Cs(2t-1)
C(s+l)(2t-1)]]_
-C(s+l)(2t_l) where Cuo is the u, v cofactor of adj(Af), then x f and
xf =
1
~_1[ Cs(2t-1)
det(A) _ _
-C~s+l)~zt-1)
_.._~1 [ D2t-I det(a) -D2t
D2t ] Dzt-1 '
Cs(2t_l) =
(Af)-113
,
(3.1)
nxn f --
det(Af) adj(af )13f,
C(s+l)(2t-1) ]13j C-s~zt-1) (3.2)
466
I". JIANG
where D2t-1 is the determinant with replaced the (2t - 1)-th column of det(A f) by the first column of f f , and D2t is the determinant with replaced the ( 2 t - 1)-th column of det(A f) by the second column of f f . Let _ -D2t
o,1:-1
_ D2t-1
6 Q,
(3.3)
then by the definition of complex representation, the quaternion linear equations Ax = f have a unique solution 1
xt = ~ A t , det(A)
t = 1, 2 . . . . . n.
(3.4)
From the statement above we get the following result. THEOREM 3.1, Let A ~ ~n×n, f ~. Qnxl. If A is an invertible quaternion matrix, then the quaternionic linear equations Ax = f have a unique solution, and the solution is 1
xt = ~ A t , det(A)
t = 1,2 . . . . . n,
where
-1
A t -- I D2t-l__ k -D2t
D2t
"O2t-1
if
E Q,
and D2t-1 is the determinant will replaced the ( 2 t - 1)-th column of det(A f) by the first column of fl y, and D2t is the determinant will replaced the (2t - 1)-th column of det(A f) by the second column of ElY. Especially, if A ~ C n×n, then by the definition of complex representation we easily know that det(A) = det(A f) = det(A)det(A), and D2t-t = Atdet(A), D2t = 0, and 1 Atdet(A) 0 if-1 Xt ~ ~ A t det(A) det(A)det(A) 0 Atdet(A) 1
1
Atdet(A) = det(A) At, det(A)det(A) in which At is the determinant with replaced the t-th column of det(A) by the quaternion vector f . Therefore the quatemlonic Cramer rule is a generalization of complex Crarner rule. 4.
C a y l e y - H a m i l t o n theorem for q u a t e r n i o n matrices
In this section we introduce a concept of characteristic polynomial of a quaternion matrix, and we give quaternionic Cayley-Hamilton theorem.
CRAMER RULE FOR QUATERNIONIC LINEAR EQUATIONS...
467
For A ~ O n×n, the characteristic polynomial of A is defined as
FAO0 =
det(~.I - A) = det(>,fI -
Af).
(4.1)
Clearly, the following result follows immediately from the Cayley-Hamilton theorem over complex field. THEOREM 4.1. Let A ~ ~nxn, and FA()Q= polynomial of A. Then FA(A) = O.
det(>,l-
A) be the characteristic
3=(
i
5. Example Let (
A=
i
1-t-j)
)
,
-1 + j
,
-k
-1
Find all solutions of the quaternionic linear equation Ax = 3. We solve the quaternionic linear equation Ax = fl in two ways: First, it is easy to find A f and flY by the definition of complex representation,
Af =
I
i 0 1 0 -i -1 -1
1
0
11 1
#
=
-I
-i
-1 -1 -i
Ii°10i
0
0
,
0
-i
and det(A) = det(A f) = 9 # 0, D1 = 6, D2 = 3, D3 .w. 3i, D4 = 0. From the quaternionic Cramer rule, the linear equation has a unique solution, and
I [ D1D21f-1 xl = det(A'--"-~ - 0 2
D1
l [ D 3 D 4 ] f-' x2 = d e ' A ) Therefore the unique solution of
- D 4 D3 /
Ax =
x =
21 = ~ d- ~j,
1. = ~t.
3 is + ~j, ~i
.
Second, it is easy to find the adjoint adj(A f) of the complex representation matrix A f, and
468
I3/° 33]E T. JIANG
adj(Af ) =
0 3 3
3i -3 3
3 0 3i
therefore the unique solution of A x x = A-lfl _ __1
det(A)
1 [-3i det(a) 3 - 3j
=
-3 3i 0
fl
=
-3i 3 - 3j
- 3 - 3j 3k
js
is
adj(A) _ _ _ 1 ( a d j ( A f ) ) f - l f l det(A) -3-3j 3k
] f l = (~ r +1 ~1 ) -'J' .,i
REFERENCES [1] D. Finkelstein, J. M. Jauch, S. Schiminovich and D. Speiser: Foundations of quaternion quantum mechanics, J. Math. Phys. 3 (1962), 207. [2] S. L. Adler: Quaternionic quantum field theory, Phys. Rev. Lett. 55 (1985), 783. [3] S. L. Adler: Quaternionic quantum field theory, Commun. Math. Phys. 104 (1986), 611. [4] S. L. Adler: Quaternionic Quantum Mechanics and Quantum Fields, Oxford University Press, New York 1995. [5] S. L. Adler and A. Millard: Coherent statex in quaternionic quantum mechanics, J. Math. Phys. 38 (1997), 2117. [6] P. Sutcliffe: Instantons and the buckyball, High Energy Physics-Theory la (2003), 157. [7] T. Jiang: An algorithm for quaternionic linear equations in quaternionic quantum theory, J. Math. Phys. 45 (2004), 4218. [8] T. Jiang: An algorithm for eigenvalues and eigenvectors of quaternion matrices in quaternionic quantum mechanics, J. Math. Phys. 45 (2004), 3334.
REPORTS ON MATHEMATICAL PHYSICS
No. 3
CRAMER RULE FOR QUATERNIONIC LINEAR EQUATIONS IN QUATERNIONIC QUANTUM THEORY* TONGSONG JIANG Department of Mathematics, Linyi Normal University, Shandong 276005, China Department of Computer Science and Technology, Shandong University, Jinan 250100, China (e-mail: [email protected]) (Received September 28, 2005)
By means of a complex representation of a quaternion matrix and a companion vector, this paper introduces a new definition of determinant for a quaternion matrix, derives a technique of finding an inverse matrix of a quaternion invertible matrix, and gives a Cramer rule for quaternionic linear equations in quaternionic quantum theory.
Keywords: Cromer rule, complex representation, companion vector, quatemionic linear equation.
1. Introduction In the study of quaternionic quantum mechanics and some other applications of quaternions [1-6], one often encounters the problem of solutions of quaternionic linear equations. Because of noncommutativity of quaternions, the solutions of quaternionic linear equations are more difficult. In papers [7, 8], by means of a complex representation and a companion vector, we have studied the problems of solutions of quatemionic linear equations and algorithms for eigenvalues and eigenvectors of a quaternion matrix, respectively, in quaternionic quantum theory. The paper [7] gave not only the necessary and sufficient conditions for the existence of solutions of quaternionic linear equations, but also a technique of finding solutions to quaternionic linear equations. This paper, by means of a complex representation and a companion vector, introduces a new definition of determinant for a quaternion matrix, gives an algorithm for inverse matrix of a quaternion invertible matrix, and gives Cramer rule for quatemionic linear equations in the quaternionic quantum theory. Let R denote the field of real numbers, C = {a + bvCL-fla, b ~ R} the field of complex numbers, and Q the field of quaternions. For x ~ C, ~ is the conjugate of x. Let IFm×n denote the set of m x n matrices on a field Y. For any A ~ C nxn, A r, det(A) and adj(A) denote the transpose, the determinant and the adjoint of the matrix A, respectively. *Supported by the National Natural Science Foundation of China and Shandong Natural Science Foundation of China (Y2005A12). [463]
464
T. JIANG
We first recall the complex representation and the companion vector [7, 8]. For any quaternion x = Xo + xli + x2j + x3k = y + z j 6 Q, in which xi 6 1R, and i 2 = j2 = k 2 = - 1 , ij = - j i = k, and a quaternion matrix A E Qm×n, the complex representations of x and A were defined respectively by
x2+x3 l[yz ]
xl = E xO+Xl
=
--X2 -I- X3"X,/'~
XO - - X l % / ~
--~
~
c2x2
(1.1)
y
and Af = (af) = [[
Yst I_1_ - g , ,
Zst ] ] ~ Y,, 13
Qf __c2m×2 n. c
(1.2)
If c~ = (xl, x2 . . . . . X2n) T ~ C 2n×1, then the companion vector otc of a vector ot was defined as oec = (-x2, xl, - x 4 , x3 . . . . . -Xzn, ~2n-1) r E C 2n×1. For any quaternion matrix A c Qm×n, by the definition of complex representation of a quaternion matrix there exist complex vectors oq, or2. . . . . oen such that A f = c c . . . , otn, otc). (Ofl,Oll,Ot2,0/2,
From [7, 8] we know that if A ~ Qm×~ and B E Qn×S, then (1.3)
(AB) f = Af B f,
and if A E Q~×n, then A is nonsingular if and and (Af) -1 = (A-l) f. Let A 6 Qn×n, then the real representation A f appear in pairs, and the imaginary conjugate pairs, respectively. The following result follows immediately from the
only if A f is nonsingular, eigenvalues of the complex eigenvalues of A f appear in statement above.
PROPOSITION 1.1. Let A ~ Qnxn and A f be its complex representation. Then the determinant det(A f) o f A f is a nonnegative real number.
2.
Determinant of quaternion matrices
In this section we define a determinant of a quatemion matrix by means of its complex representation and companion vector. DEFINITION 2.1. Let A ~ Qn×n and A f be its complex representation. Then determinant det(A) of the quaternion matrix A is defined as det(A) = det(Af).
(2.1)
From this definition and Proposition 1.1 we easily know that the determinant of a quaternion matrix is a nonnegative real number, and if A c Qn×n, B ~ Q~×n, then det(AB) = det(A) det(B).
(2.2)
Let A c Qn×n. Then it is easy to show by a direct calculation that adj(A f) E Q f . Now we introduce the concept of adjoint matrix of a quaternion matrix A by means of complex representation.
CRAMER RULE FOR QUATERNIONIC LINEAR EQUATIONS... D E F I N I T I O N 2.2. Let A c Qn×, and adjoint adj(A) of A is defined as
Af
be its complex representation. Then the
adj(A) = (adj(Af )) f-~. For any A
~
465
(2.3)
Qnxn we have (2.4)
(A adj(A)) / = A f (adj(A)) f = det(Af )I2,,
and by the definitions of complex representation and determinant of a quaternion matrix, A adj(a) = det(af)I n = det(A)In. Similarly, adj(A)A = det(A)In. PROPOSITION 2.1. Let A ~ (~×~, then (1) A adj(a) = a d j ( a ) a = det(a)In; (2) A is an invertible quaternion matrix if and only if det(A) # 0, in which case 1 A -1 -- - adj(A). (2.5) det(A) Proposition 2.1 not only gives the necessary and sufficient conditions for a quaternion matrix to be invertible, but also gives a technique of finding the inverse by means of the complex representation of the matrix.
3.
Cramer rule for quaternionic linear equations In this section we give Cramer rule for quaternionic linear equations in quaternionic quantum mechanics and quantum fields by means of complex representation of the quaternion matrix. If A 6 Qn×n, 13 6 (~n×l, then by the definition of complex representation, Ax = 13 if and only if A f x f = 13f. That is, Ax = 13 has a solution x if and only if A f Y = 13f has a solution Y = x f. If a quaternion matrix A is invertible, i.e. det(A) = det(A f) # 0, then the complex linear equations A f Y = 13f have a unique solution x f. Since adj(A f) E Qf, so we may put adj(Af)=[[
_Cs(2t-1)
C(s+l)(2t-1)]]_
-C(s+l)(2t_l) where Cuo is the u, v cofactor of adj(Af), then x f and
xf =
1
~_1[ Cs(2t-1)
det(A) _ _
-C~s+l)~zt-1)
_.._~1 [ D2t-I det(a) -D2t
D2t ] Dzt-1 '
Cs(2t_l) =
(Af)-113
,
(3.1)
nxn f --
det(Af) adj(af )13f,
C(s+l)(2t-1) ]13j C-s~zt-1) (3.2)
466
I". JIANG
where D2t-1 is the determinant with replaced the (2t - 1)-th column of det(A f) by the first column of f f , and D2t is the determinant with replaced the ( 2 t - 1)-th column of det(A f) by the second column of f f . Let _ -D2t
o,1:-1
_ D2t-1
6 Q,
(3.3)
then by the definition of complex representation, the quaternion linear equations Ax = f have a unique solution 1
xt = ~ A t , det(A)
t = 1, 2 . . . . . n.
(3.4)
From the statement above we get the following result. THEOREM 3.1, Let A ~ ~n×n, f ~. Qnxl. If A is an invertible quaternion matrix, then the quaternionic linear equations Ax = f have a unique solution, and the solution is 1
xt = ~ A t , det(A)
t = 1,2 . . . . . n,
where
-1
A t -- I D2t-l__ k -D2t
D2t
"O2t-1
if
E Q,
and D2t-1 is the determinant will replaced the ( 2 t - 1)-th column of det(A f) by the first column of fl y, and D2t is the determinant will replaced the (2t - 1)-th column of det(A f) by the second column of ElY. Especially, if A ~ C n×n, then by the definition of complex representation we easily know that det(A) = det(A f) = det(A)det(A), and D2t-t = Atdet(A), D2t = 0, and 1 Atdet(A) 0 if-1 Xt ~ ~ A t det(A) det(A)det(A) 0 Atdet(A) 1
1
Atdet(A) = det(A) At, det(A)det(A) in which At is the determinant with replaced the t-th column of det(A) by the quaternion vector f . Therefore the quatemlonic Cramer rule is a generalization of complex Crarner rule. 4.
C a y l e y - H a m i l t o n theorem for q u a t e r n i o n matrices
In this section we introduce a concept of characteristic polynomial of a quaternion matrix, and we give quaternionic Cayley-Hamilton theorem.
CRAMER RULE FOR QUATERNIONIC LINEAR EQUATIONS...
467
For A ~ O n×n, the characteristic polynomial of A is defined as
FAO0 =
det(~.I - A) = det(>,fI -
Af).
(4.1)
Clearly, the following result follows immediately from the Cayley-Hamilton theorem over complex field. THEOREM 4.1. Let A ~ ~nxn, and FA()Q= polynomial of A. Then FA(A) = O.
det(>,l-
A) be the characteristic
3=(
i
5. Example Let (
A=
i
1-t-j)
)
,
-1 + j
,
-k
-1
Find all solutions of the quaternionic linear equation Ax = 3. We solve the quaternionic linear equation Ax = fl in two ways: First, it is easy to find A f and flY by the definition of complex representation,
Af =
I
i 0 1 0 -i -1 -1
1
0
11 1
#
=
-I
-i
-1 -1 -i
Ii°10i
0
0
,
0
-i
and det(A) = det(A f) = 9 # 0, D1 = 6, D2 = 3, D3 .w. 3i, D4 = 0. From the quaternionic Cramer rule, the linear equation has a unique solution, and
I [ D1D21f-1 xl = det(A'--"-~ - 0 2
D1
l [ D 3 D 4 ] f-' x2 = d e ' A ) Therefore the unique solution of
- D 4 D3 /
Ax =
x =
21 = ~ d- ~j,
1. = ~t.
3 is + ~j, ~i
.
Second, it is easy to find the adjoint adj(A f) of the complex representation matrix A f, and
468
I3/° 33]E T. JIANG
adj(Af ) =
0 3 3
3i -3 3
3 0 3i
therefore the unique solution of A x x = A-lfl _ __1
det(A)
1 [-3i det(a) 3 - 3j
=
-3 3i 0
fl
=
-3i 3 - 3j
- 3 - 3j 3k
js
is
adj(A) _ _ _ 1 ( a d j ( A f ) ) f - l f l det(A) -3-3j 3k
] f l = (~ r +1 ~1 ) -'J' .,i
REFERENCES [1] D. Finkelstein, J. M. Jauch, S. Schiminovich and D. Speiser: Foundations of quaternion quantum mechanics, J. Math. Phys. 3 (1962), 207. [2] S. L. Adler: Quaternionic quantum field theory, Phys. Rev. Lett. 55 (1985), 783. [3] S. L. Adler: Quaternionic quantum field theory, Commun. Math. Phys. 104 (1986), 611. [4] S. L. Adler: Quaternionic Quantum Mechanics and Quantum Fields, Oxford University Press, New York 1995. [5] S. L. Adler and A. Millard: Coherent statex in quaternionic quantum mechanics, J. Math. Phys. 38 (1997), 2117. [6] P. Sutcliffe: Instantons and the buckyball, High Energy Physics-Theory la (2003), 157. [7] T. Jiang: An algorithm for quaternionic linear equations in quaternionic quantum theory, J. Math. Phys. 45 (2004), 4218. [8] T. Jiang: An algorithm for eigenvalues and eigenvectors of quaternion matrices in quaternionic quantum mechanics, J. Math. Phys. 45 (2004), 3334.