Normal form of skew-symmetric matrix

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Normal form of skew-symmetric matrix

Makito Oi Institute of Natural Sciences, Senshu University

Abstract. A discussion on the factorisation of real skew-symmetric matrices is presented. Its relation with physical contexts is briefly considered.

1. Introduction

1.1 Quadratic form and symmetric matrices

The spectral theory is originated from D. Hilbert’s interest on the mathematical nature of the quadratic form, which has a geometric connection to the search of the principal axes of an ellipse (2D) or an ellipsoid (3D).

In the 2-dimensional case, the general form of the quadratic form is given as ax2+2bxy + cy2,

where (x, y) and (a, b, c) are real variables and real constants, respectively. In terms of matrices and vectors, the quadratic form can be expressed as

ax2+2bxy + cy2vTSv, (1) where v   x y , vT ( x y ) , (2) and S   a b b c . (3)

The most significant characteristics about the quadratic form is that the matrix S has a form of a symmetric matrix, or ST S.

The equation of an ellipse is given as vTSv const. Its principal axes can be found by

“diago-nalising” the symmetric matrix. In mathematics, the diagonalised form is also referred to as the “normal form” or “canonical form”. In the present case, S is decomposed to

S  OΛOT, (4)

where O is an orthogonal matrix OOT OTO  Iand Λ is a diagonal matrix,

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The decomposition expression can be rearranged to

SO  OΛ, (6)

and O can be expressed with the orthonormal basis vias

O (v1, v2). (7)

In this way, the decomposition has an alternative form as known as the eigenvalue equation,

Sviλivi, (8)

where i  1, 2. λiare called “eigenvalues” and the associated vector viis termed as “eigenvector”.

In physics, the diagonalisation problem is essential, especially in quantum mechanics, where all the dynamical variables are represented in terms of Hermitian matrices H  H†. Observations

of a dynamical variable are given through the relevant eigenvalues. The Hermitian matrices are reduced to symmetric matrices when they are real matrices, so that the study of the quadratic form is important in physics.

1.2 An extention of quadratic form with skew-symmetric matrix

The well-known quadratic form with a symmetric matrix can be extended to the bi-linear form with a skew-symmetric matrix A  −AT, which is similar to commutation relations in quantum

mechanics and external product (or cross product) seen, for example, in electromagnetism and quantum mechanics.

The normal form of such a bi-linear form is defined, in the 2-dimensional case, as x1y2− x2y1,

which can be expressed as

x1y2− x2y1xTJy, (9) where J    0 1 −1 0 . (10)

When the vectors are chosen as a pair of canonical variables xT  (x, p)  yT in the (classical

and quantum) mechanics, then the canonical form of the bi-linear expression corresponds to the commutator [x, p]  xp−px in quantum mechanics and to the Poisson bracket in classical mechanics. In the 2n-dimensional case, the general expression for the bi-linear form is given as xTAy, but

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general expression can be converted into the normal form. x and y are transformed by P as

x  P x′, y  P y′. (11)

The bi-linear form is then transformed as

xTAy  x′TPTAP y′. (12)

When PTAP is a skew-symmetric matrix A  −A′T, the bi-linear form is conserved by the

transformation P , and A and A′is said to be “similar”. By repeating this similar transformations,

one can finally reach the form of J. There is a theorem that such a transformation can be summarised by a single similar transformation P [1]. This statement is equivalent to a fact that A can be decomposed to A  P JPT, (13) where J is redefined to J diag      0 1 −1 0 ,   0 1 −1 0 , · · ·   . (14)

When P is restricted to an orthogonal matrix PTP  P PT I, there is a theorem [2] that the

decomposition of A is given as A  P ΛPT, (15) where Λ  diag      0 λ1 −λ1 0   ,   0 λ2 −λ2 0   , · · ·   . (16)

λiare real and positive numbers (λi>0) when A is a full-rank skew-symmetric matrix. The above

expression is equivalent to the following form

AP  P J, (17)

and it is further rewritten as a set of pairwise equations

Avi  −λiui, (18)

Aui  λivi, (19)

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if P is expressed as a collection of orthogonal basis {vi, ui},

P (v1, u1, v2, u2,· · ·). (21) The final expression looks like an eigenvalue equation and λiseems to be related to the relevant

eigenvalues. The aim of this short paper is to investigate whether this conjunction is true or false. 2. Basic properties of real skew-symmetric matrix

For the sake of simplicity, we consider in this paper only the case of real skew-symmetric matrices A −AT, A  A. First of all, I would like to summarise the well-known properties associated with the eigenvalue problem of A,

Aviαivi. (22)

Unlike the quadratic-form theory (as well as its derivative bi-linear form), the inner product must be defined “properly”, that is, complex conjugate must be attached to the transpose operation 1. This is because the eigenvectors viof A involve complex elements. For instance, let us consider the

simplest case of the 2-dimensional problem, A  J    0 1 −1 0 . (23)

The eigenvalues are λ1iand λ2−i, and their eigenvectors are respectively v1 √21 (1, i)T and

v2 √21 (1, −i)T . Therefore, to secure the positive-definiteness of the norm, we must define the inner product ⟨v|u⟩ between v and u as

⟨v|u⟩  v∗T · u 

i

v∗iui. (24)

Following Dirac’s notation, let us write the eigenvalue equation as

A|vi⟩  αi|vi⟩. (25)

With a replacement of index i with j, the dual vector space is defined so as to be consistent with the inner product through

⟨vj|A†⟨vj|(αj), (26)

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where the hermitian conjugate † is given as H†H∗T. Using the above two equations (by applying

⟨vj| from left in the former while |vi⟩ from right in the latter), we have two relations

⟨vj|A|vi⟩  αi⟨vj|vi⟩, (27)

⟨vj|A†|vi⟩  (αj)⟨vj|vi⟩. (28) In the present case, we are dealing with the real skew-symmetric matrix, so that A† −A. Then, the above equations can be reduced to

{

αi+(αj)}⟨vj|vi⟩  0. (29)

When i  j, we have

αi−(αi), (30)

which means the eigenvalues are pure imaginary αii, where λiare real numbers. Putting this

result in Eq.(29), we have

(λi− λj)⟨vj|vi⟩  0, (31)

which means the eigenvectors are orthogonal when their eigenvalues differ.

Using the reality of A, we can also demonstrate that −iλiis an eigenvalue if iλiis an eigenvalue.

Taking the complex conjugate of the eigenvalue equation, we have

A∗vi∗−iλivi∗. (32)

Because A∗A, the above equation is nothing but the eigenvalue equation for the eigenvalue −iλi, with the associated eigenvector being vi∗. Because of this property,we have

⟨v∗i|vi⟩ 

k

(vi)k20, (33)

which simply implies that the vector components are complex numbers. 3. Eigenvalue decomposition

Using the above properties, the eigenvalue equation for A is expressed as a decomposition form,

AV  V Λ, (34)

where

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and

Λ  diag(1,−iλ1, iλ2,−iλ2,· · ·)i· diag(λ1,−λ1, λ2,−λ2,· · ·). (36) With a proper normalisation of the vectors vi, V becomes a unitary matrix, VV  V V  I. Therefore, we have

A  V ΛV†. (37)

4. Normal form

Because A is a skew-symmetric matrix, A can be also decomposed with the normal form. The starting point is the eigenvalue equation:

Aviivi, (38)

and

Aui−iλiui, (39)

where

ui(vi). (40)

In order to obtain the normal form, an alternating relation between vi and uiis required. Using

the above complex conjugate relation, the first two eigenvalue equations are rewritten to

Avi  i(ui), (41)

Aui  −iλi(vi). (42)

These two equations can be collectively expressed as

A(v1u1v2u2 · · ·)  (1u1∗, −iλ1v1∗, iλ2u2∗, −iλ2v2∗, · · ·) (43)  −i(−λ1u1∗, λ1v1∗, −λ2u2∗, λ2v2∗,· · ·) (44)  −i(v1u1v2u2 · · ·)diag      0 λ1 −λ1 0   ,   0 λ2 −λ2 0   , · · ·   . (45) Using the fact that(v1u1v2u2 · · ·) is a unitary matrix, A can be decomposed to

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where U is a unitary matrix, whose representation is given, U exp(−iπ

4 ) (

v1u1v2u2 · · ·)∗, (48)

and the normal matrix N is N diag      0 λ1 −λ1 0   ,   0 λ2 −λ2 0   , · · ·   . (49)

4.1 The case of 2-dimension

In the 2-dimensional case, there is the only one skew-symmetric matrix essentially, which is J    0 1 −1 0 . (50)

As already obtain earlier, the eigenvalues of J are ±i with the associated eigenvectors being (

1, ±i)T/√2, so that the normal-form decomposition becomes   0 1 −1 0   exp ( −iπ4) √2   1 1 −i i     0 1 −1 0    1 i 1 −i  exp ( −iπ4) √2 , (51) or J  U JUT, (52) where U exp ( −iπ 4) √2   1 1 −i i   exp ( −iπ 4) √2  1 1 i −i   . (53) It is easy to confirm U†U  U UI. 5. Applications of the decompositions

To evaluate the determinant of an 2n-by2n real skew-symmetric matrix A, the eigenvalue decom-position is useful.

detA  det(U ΛU†)det(UU†)detΛ 

nj (iλj)(−iλj)  nj λ2j. (54)

To evaluate the Pfaffian of A, the normal-form decomposition is useful. PfA  Pf(UNUT)  detU · PfN  exp(iγ)

n

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Here, we write detU  exp(iγ), where γ is an arbitrary real number. By this choice, the unitary condition

det(UU†)  |detU|21 (56)

is satisfied. However, we should note that the definition of the Pfaffian is given as PfA  2n1n!σ∈S2n sgn(σ) ni1 aσ(2i−1),σ(2i), (57)

where S2nis the symmetry group of degree (2n) and σ is a permutation of (1, 2, · · · , 2n) and sgn(σ)

is the corresponding sign. The above definition means that the Pfaffian of a real skew-symmetric matrix A should be a real number because aij∈ R. Therefore, γ is required to be kπ, where k is an

integer, or detU  (−1)k.

In the 2-dimensional case, detU  det  exp ( −iπ 4) √2   1 1 −i i      ( exp(−iπ 4) √2 )2 2i  1, (58)

so that k is chosen to be an even integer. 6. Numerical experiments

As for the higher dimensional cases (n ≥ 4), numerical calculations are carried out for examining the value of k in detU  (−1)k. The programming language is chosen to be FORTRAN (the

compiler is gfortran, ver. 6.3.0), and the LAPACK subroutine library is used in the programme. The diagonalisation is performed with zgeev, which allows complex matrices without any symmetries. 6.1 n  4

The first example is the following real skew-symmetric matrix,

A           0 2 1 −1.2 −2 0 5.23 −1.23d − 3 −1 −5.23 0 5.23d − 2 1.2 1.23d − 3 −5.23d − 2 0          . (59)

The resultant normal form is

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and the unitary matrix is U e−iπ4         

0.12004 − 0.26343i 0.12004 + 0.26343i 0.00113 + 0.64513i 0.00113 − 0.64513i

0.69262 0.69262 0.06035 − i0.12898i 0.06035 + 0.12898i

0.04589 + 0.65573i 0.04589 − 0.65573i 0.02723 + 0.25917i 0.02723 − 0.25917i −0.06134 − 0.02495i −0.06134 + 0.02495i 0.70400 0.70400          . (61) The unitarity of U is numerically confirmed, although its detail is omitted here. The determinant of U is evaluated to

detU  −0.99999999999999911 − (3.8857805861880479E − 016)i, (62) that is, detU  −1 in this example.

Next, we change the sign of the matrix elements of a14−1.2 and a24−1.23d − 3 to a141.2 and a24  1.23d − 3, respectively. The resultant eigenvalues are ±5.70484i and ±1.11824i,and the determinant of the associated unitary matrix U′ becomes detU  1.0000000000000009 + (2.7755575615628914E − 016)i ≃ 1. The sign of the determinant is now inverted in comparison with the previous example.

6.2 n=6

The first example is

A                 0 2 1 −1.2 1.23 −220 −2 0 5.23 −1.23d − 3 520 −4.31d − 3 −1 −5.23 0 5.23d − 2 8.15d − 1 −77.5 1.2 1.23d − 3 −5.23d − 2 0 −0.992 −45.2 −1.23 −520 −8.15d − 1 0.992 0 −6.81 220 4.31d − 3 77.5 45.2 6.81 0                . (63)

The resultant normal form is N diag      0 520.08873 −520.08873 0   ,   0 237.83573 −237.83573 0   ,   0 4.13111 −4.13111 0     , (64) and the determinant of the corresponding unitary matrix U is

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When we change the sign of the a14  −1.2 matrix element to a14  1.2, then the determi-nant of the relevant unitary matrix changes to detU  1 ( which is numerically evaluated to be 0.99999999999999734 + (2.7200464103316335E − 015)i).

6.3 A hypothesis

It seems that the value of k in detU  (−1)kis nothing to do with the dimension of the matrix A,

as demonstrated in the above numerical calculations. 7. Conclusion

Two ways to decompose real skew-symmetric matrix A are presented: (i) eigenvalue decom-position and (ii) normal-form decomdecom-position. These decomdecom-positions are useful to evaluate the determinant and the Pfaffian of A. To examine the validity of the obtained decomposition methods, some numerical calculations are performed for confirmation.

Acknowledgement

This research is supported by a Senshu-University grant (Kenkyu Josei) for the study of “Ryoshi tatai-kei ni okeru tsui-sokan oyobi koji-sokan no suri-teki katsu suchi-teki toriatukai ni tsuite no kenkyu” (H.31), which is appreciated by the author.

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