Mental Properties in Linear Optics

S Afr Optom 2010 69(1) 3-13 Symplecticity and relationships among the fundamental properties in linear optics

WF Harris

Department of Optometry, University of Johannesburg, PO Box 524, Auckland Park, 2006 South Africa

Received 4 January 2010; revised version accepted 5 March 2010

Abstract the basic notions underlying symplecticity. The relationship to another important class of matrices, Because of symplecticity the four fundamental the Hamiltonian matrices, is discussed together first-order optical properties of an optical system with their potential role in statistical analysis of are not independent. Relationships among them the eye. Augmented symplectic matrices are also reduce the number of degrees of freedom of a sys- defined and their relationship to augmented Hamil- tem’s transference from 16 to 10. There are many tonian matrices described. An appendix gives nu- such relationships, they are not easy to remember, merical examples of symplectic and Hamiltonian they take many forms and they are often needed in matrices and shows how they may be recognized derivations. The purpose of this paper is to provide and constructed. (S Afr Optom 2010 69(1) 3-13) in one place a comprehensive collection of those that have proved useful in linear optics generally Key words: symplecticity, Schur complement, and in the context of the eye particularly. The paper symmetric product, Hamiltonian matrix, augment- also offers aids to memorizing some of the results, ed symplectic matrix, augmented Hamiltonian ma- derives most of them and along the way introduces trix

Introduction relationships can take many unfamiliar forms, are not easy to remember, and are often needed in analyses of Symplecticity is of profound significance to mod- optical problems it would seem useful to have a com- ern science and to optometry in particular. It can even pact and more accessible summary. Accordingly the be argued (see towards the end of this paper) that it is objective of this paper is to supply such a summary. the reason why refractive errors can be compensated by means of conventional spherocylindrical lenses Basic results of linear algebra and is, therefore, no less than a sine qua non of optometry. We make use of the basic results of linear algebra Symplecticity implies particular relationships as presented in introductory texts5-8. Our matrices are among the fundamental optical properties of an opti- all real, that is, their entries are all real numbers. In cal system. Although there are several good sources particular for square matrices A and B that deal with symplecticity1-4 they tend to be math- (ABT ) = BTAT (1) ematically sophisticated and not readily accessible for where AT is the matrix transpose of A. Also if most people working in visual optics. Because the AB= BA = I , (2)

3 The South African Optometrist S Afr Optom 2010 69(1) 3-13 WF Harris - Symplecticity and relationships among the fundamental properties in linear optics where I is an identity matrix, then B is the inverse of I and O are n × n identity and null matrices respec- A and is written A−1 . Then tively. In general n is any positive integer although _ _ _ (AB 1) = B 1A 1, (3) only n = 1 and n = 2 are needed on the optical con- provided the inverses exist. Also text; n = 1 applies in Gaussian optics (the simplest _ _ (A 1)T = (AT) 1. (4) optics in the plane) and n = 2 applies in linear op- _ We use the common abbreviation A T for either side tics (the simplest optics in three dimensions and the of Equation 4. Also simplest proper approach to astigmatism). E itself is det(AB)=det A det B (5) 2n × 2n . It is sometimes called the symplectic unit and matrix10. Although symbols vary, more often than not T det A =det A . (6) it is represented by J. (We use E instead because J is Partitioned matrices feature importantly in sym- already used for one of the basic matrices in the set I, plecticity. They take the form J, K and L.)  A B Applying Equation 8 we find that S =   , (7) 2 C D E = −I . (12) (Notice that I does not have the same meaning in a matrix of matrices as it were. They are no differ- Equations 11 and 12: in the former it is n × n , in the ent from ordinary matrices, however, and obey the usual rules of matrix algebra. Equation 7 does not latter 2n × 2n .) Hence imply that S is 2× 2 though it is 2× 2 if each of the E(−E)= (−E)EE==II . (13) submatrices A, B, C and D is 1×1. S is 4× 4 if each It follows from Equations 2 and 13 that of the submatrices is 2× 2 . In optical applications S E−1 = −E . (14) represents the transference of an optical system and It follows from Equation 10 that T the submatrices the ( 2× 2 ) fundamental linear opti- E = −E . (15) 5-8 cal properties of the system. From the definition of the determinant it turns Multiplication takes the form out that      det E = 1. (16) A BE  F   AE+ BG AF + BH   =   (8)     + +  C D  G H   CE DG CF DH Symplectic matrices as might be expected. Multiplication by a scalar is as By definition a matrixS is symplectic if expected: ABAB  s s  STES=E. (17) s=   . (9) CDCD  s s  The transference of an optical system is symplectic1, 4. The transposition operator is taken inside the par- Different optical systems can have the same transfer- titioned matrix but the off-diagonal submatrices are ence. For every 2× 2 or 4× 4 symplectic matrix it is also interchanged: possible to have an optical system whose transference T T T  A B  A C  is that matrix.11, 12   =   . (10) C D  B T DT  Suppose S1 and S2 are both symplectic (and have It might be supposed that similar simple expres- the same n). Consider the product S1S2 . Substitut- sions can be written for the inverse and determinant ing it into the left-hand side of Equation 17 we ob- of a partitioned matrix; that, however, is not the case tain in general. (There are expressions but they are much T 9 S S E S S more complicated .) ( 1 2 ) ( 1 2 ) T TT T The symplectic unit matrix = =SS2 S21SES1 ESE1S1S2by2 Equation 1 T = S2 ES22 because S1 is symplectic Consider the partitioned matrix = E because S2 is symplectic.  O I  In other words S S satisfies Equation 17 and so is E :=   . (11) 1 2  −I O

4 The South African Optometrist S Afr Optom 2010 69(1) 3-13 WF Harris - Symplecticity and relationships among the fundamental properties in linear optics symplectic. This means that the product of symplec- obtains tic matrices is symplectic or, in other words, symplec- T T T (22) tic matrices are closed under multiplication. E= (S ) ES On the other hand symplectic matrices are not in which shows that ST is symplectic. general closed under addition or multiplication by Let symplectic matrix S be partitioned as in Equa- a scalar. (The reader is encouraged to show this by tion 7. Then, from Equation 10, working out some simple examples.) This means,  AT CT  in particular, that an arithmetic average of symplec- ST =   . (23) tic matrices is not in general symplectic, a fact that  T T   B D  has important implications for basic statistics. If one Substitution into Equation 17 results in wants to calculate an average eye, for example, one − CT A + ATC − CTB + ATD  O I  would want the average to be a possible eye. The   =   . (24)  T T T T    fact that the arithmetic average of transferences is not  − D A + B C − D B + B D − I O symplectic in general implies that, strictly speaking, Equating the four blocks on the left and right we see it is not meaningful to calculate an average eye that that way. The problem of how to calculate an average eye T T is not a simple one. The relationship between sym- A C = C A , (25) plectic and Hamiltonian matrices, to be discussed be- BTD = DTB , (26) low, appears to offer a solution13-20. Despite what we ATD − CTB = I have said here the arithmetic average may, in some (27) cases, be sufficiently close to being symplectic for it and to be a good enough approximation. DT A − BTC = I . (28) It is easy to see that I and E are themselves symplectic: they satisfy Equation 17. On the other hand Transposition applied to Equation 28 shows that it is O does not and, therefore, is not symplectic. equivalent to Equation 27. Making use of Equations 6, 16 and 17 we find When S is 2× 2 Equations 25 and 26 are trivially that true and Equation 28 reduces to Equation 20. If the (det S)2 = 1. (18) four entries of a 2× 2 matrix are chosen arbitrarily then the matrix is usually not symplectic. To con- This suggests that struct a 2× 2 symplectic matrix one is free to choose at most three of the entries arbitrarily; the fourth is de- det S = ±1. (19) termined by Equation 20. One can say that symplec- In fact it turns out that ticity implies a loss of one degree of freedom from four to three. Note that Equation 20 implies that a det S = 1. (20) (The proof is not simple and will not be attempted 2× 2 symplectic matrix cannot have a row or a col- here. Proofs are given elsewhere2-4.) This means umn of zeros. a symplectic matrix S is never singular (that is, When S is 4× 4 Equations 25 and 26 imply a −1 loss of one degree of freedom each and Equation 27 det S ≠ 0 ) and its inverse S always exists. a loss of four degrees of freedom. Thus, instead of −T Premutiplying both sides of Equation 17 by S 16 degrees of freedom, a 4× 4 symplectic matrix has and postmultiplying by S−1 one obtains only 10 degrees of freedom. Constructing a 4× 4 _ _ 1 T 1 E= (S ) ES . (21) symplectic matrix, however, is not easy. A couple of methods will be described later. Comparing this with Equation 17 we see that, if S is Because ST is symplectic Equation 22 shows that −1 symplectic, then so is S . TT TT −BA + AB − BC + AD  OI  Inverting both sides of Equation 17 and pre-  =  . (29)  TT TT    −DA + CB − DC + CD  −IO  multiplying by S and postmultiplying by S−1 one

5 The South African Optometrist S Afr Optom 2010 69(1) 3-13 WF Harris - Symplecticity and relationships among the fundamental properties in linear optics

Hence tiplying by A−T results in ABT = BAT , (30) −1 −T T CA = A C (35) CDT = DCT , (31) provided A is nonsingular. Hence ADT _ BCT = I (32) _ _ CA 1=(CA 1)T (36) and _ showing that CA 1 is symmetric. In the same way DAT _ CBT = I . (33) one finds other products that are symmetric. Thus

_ _ _ _ Equation 33 is equivalent to Equation 32. AC 1, CA 1, BD 1, DB 1 In order to test whether a given matrix is symplec- _ _ _ _ tic or not one can proceed as follows. One can direct- A 1B, B 1A, C 1D, D 1C, ly test whether it obeys Equation 17. Alternatively one can test whether all three of Equations 25 to 27 are all necessarily symmetric. One might say ‘rows, (or Equations 30 to 32) are obeyed. In either case if inverse first; columns, inverse second’. The initial _ the matrix does obey the equation or equations then it letters spell rifcis. Other products, including A 1C, _ _ _ is symplectic; if it does not then it is not symplectic. B 1D, AB 1, CD 1 are not generally symmetric. Numerical examples are presented in the Appendix. For an eye B−1A is the corneal-plane refractive compensation21 and A−1B is the optical structure (a Symmetric products point or interval of Sturm) conjugate to the retina22. _ _ The negatives of CA 1 and D 1C are back- and front- Applying Equation 1 to the right-hand side of vertex powers of an optical system23. Equation 25 we obtain

ATC = (ATC)T . (34) The inverted symplectic matrix In other words ATC is symmetric. The same approach Because of Equations 25 to 28 multiplication can be applied to Equations 26, 30 and 31. One reach- shows that es the conclusion that, although A, B, C and D in S (Equation 7) can be symmetric or asymmetric, all of  T T  A B  D −B  the following products are necessarily symmetric:   = I (37)  T T   T T T T −C A C D A C, C A , B D , D B , for a symplectic matrix S partitioned as in Equation T T T T AB , BA , CD , DC . 7. Similarly, because of Equations 30 to 32, one finds that Notice that, in these cases, the two submatrices in a T T product occupy the same row or the same column of  A B D −B     = I . (38) S. Furthermore, if the two submatrices share the same    C D −CT AT column of S then the first of the two is transposed; if    the two are in the same row then the second is trans- By Equation 2 then we see that posed. As an aid to memory one might say ‘rows, transpose second; columns, transpose first’. Other T T T T T  D −B  products, including AC, BD , A B, C B, AB, BC, S−1 =   . (39)  T T  etc., are not symmetric in general. −C A  Postmultiplying Equation 25 by A−1 and premul-

6 The South African Optometrist S Afr Optom 2010 69(1) 3-13 WF Harris - Symplecticity and relationships among the fundamental properties in linear optics for any symplectic matrix S. Thus the inverse of the sequence of submatrices on the left-hand sides of a symplectic matrix is what one might expect for Equations 40 to 43 the following may be helpful: be- a 2× 2 matrix except that the submatrices are also  A B ginning diagonally across in   from the sub- transposed.   C D The Schur complements matrix in question one writes the submatrices in cy- clical order going clockwise for the block-diagonal For A, B, C and D the submatrices of a matrix S as matrices and anticlockwise for the others. in Equation 7 the expression A − D−TBTC is known as the Schur complement of A in S. Similarly there Hamiltonian matrices are Schur complements of B, C and D in S. Schur complements arose out of the work of I Schur24 and There is another class of matrices which is impor- are of considerable modern scientific interest25. It is tant in modern science and which bears a surprising no surprise that they should arise in visual optics. relationship to symplectic matrices: it is the class of Hamiltonian matrices. A matrix H is Hamiltonian if If S is symplectic then the Schur complements re- 10, 30-34 duce to particularly neat expressions: it obeys HTE = ETH . (46) _ _1 _T A BD C = D , (40) E remains the symplectic unit matrix defined by _ _ B _ AC 1D = _C T, (41) Equation 11 above. In general H is 2n × 2n but, as with symplectic matrices, we are interested only in _ _1 _ _T C DB A = B , (42) Hamiltonian matrices with n = 1 (Gaussian optics) or n = 2 (linear optics). _ _1 _T D CA B = A . (43) It is a remarkable fact that the principal matrix logarithm of a symplectic matrix is a Hamiltonian matrix These results were apparently first obtained by Dopi- and the matrix exponential of a Hamiltonian matrix is 26 co and Johnson . In connection with eyes they have symplectic10, 17, 32-34. The exponential of a real square 22, 27-29 been involved in several papers although not al- matrix X is the real infinite convergent series ways recognised as such. To prove Equation 40 we premultiply each side of ∞ _T 1 Equation 28 by D to give eX := ∑ X j . (47) j=0 j! A − D−TBTC = D−T (44) Any matrix X which satisfies and apply Equation 1 to give X e = A (48) _ _1 T _T A (BD ) C= A . (45)

_ is called a logarithm of A; in general there is more Equation 40 follows because BD 1 is symmetric. than one. However, if none of the eigenvalues of A Equations 41 to 43 follow similarly. is zero or a negative real number, then A does have a Thus the Schur complements of the block-diago- unique real logarithm the magnitude of whose eigen- nal submatrices of transference S are the transposed values are less than π . It is the principal logarithm inverses of their opposites while the Schur comple- and is written LogA . So, if S is symplectic then ments of the other submatrices are the negatives of = the transposed inverses of their opposites. To recall LogS H (49)

7 The South African Optometrist S Afr Optom 2010 69(1) 3-13 WF Harris - Symplecticity and relationships among the fundamental properties in linear optics

is Hamiltonian and if H is Hamiltonian then Hamiltonian but that I is not. Suppose H is partitioned as H e = S (50) M N H =   . (55)   is symplectic.  P Q It is important to note that these are not the famil- iar logarithm and exponential simply applied to the If H is Hamiltonian then according to Equations 46 entries of the matrix separately. (In Matlab they are and 11 T T given by the functions logm and expm as opposed to M N  O I   O I  M N log and exp which operate separately on the entries of     =             (56) the matrix.) See Example 3 in the Appendix.  P Q −I O −I O  P Q Let H be a Hamiltonian matrix and s a scalar. It is easy to show that the matrix sH obeys Equation 46 which multiplies out to and so is Hamiltonian. (The left-hands side of Equa-  −PT MT  −P −Q tion 46 equals sHTE and the right-hand side equals   =   . (57)  T T    T −Q N   M N  sE H but the two sides are equal because of Equa- tion 46.) Hence Hamiltonian matrices are closed un- This shows that der multiplication by a scalar. PT = P , (58) Now suppose H1 and H 2 are each Hamiltonian T and have the same n. Then Equation 46 applies to N = N (59) each: and

T . (60) T T Q = −M H1 E = E H1 (51) In other words, for a Hamiltonian matrix partitioned and as in Equation 55, the off-diagonal submatrices N and P are necessarily symmetric and one diagonal subma- T T H 2 E = E H 2 . (52) trix is the negative of the transpose of the other. For a 2× 2 Hamiltonian matrix Equations 58 and Addition leads to 59 are trivially true and Equation 60 is equivalent to the statement that T TTT TT TT T TTTT (53) trH = 0 (61) (H1 ((H+(HH1H11+2+)HHE22=)))EEEEE===(=EHEEE1 (+(H(H(HH111+12++)+HHHH222)2))) or, in other words, that a Hamiltonian matrix has zero trace or is traceless as it is sometimes called. Thus, and so as in the case of symplectic matrices, there is a loss of one degree of freedom from four to three. In fact the T T (H1 + H 2 ) E = E (H1 + H 2 ) (54) 2× 2 Hamiltonian matrices define a three-dimensional vector or linear space which means that it is possible to draw three-dimensional graphs representing the which shows that H1 + H 2 obeys Equation 46. Hence Hamiltonian matrices are closed under addition. space. Individual matrices can be plotted in the space That Hamiltonian matrices are closed under ad- to form trajectories and clusters. An arithmetic mean dition and multiplication by a scalar means that the is a point in the same space at the centre of a cluster. arithmetic average of Hamiltonian matrices is itself 3× 3 variance-covariance matrices provide measures Hamiltonian. of spread and variation in the space. It is apparent from Equation 46 that O and E are For 4× 4 Hamiltonian matrices Equations 58 and

8 The South African Optometrist S Afr Optom 2010 69(1) 3-13 WF Harris - Symplecticity and relationships among the fundamental properties in linear optics

59 each represent a loss of a degree of freedom and  I B Equation 60 a loss of four degrees of freedom. Thus,   form   where B is symmetric but an optical also as for symplectic matrices, there is a loss of six O I  degrees of freedom from 16 to 10. The 4× 4 Hamil- system with this transference is not simple.) tonian matrices define a 10-dimensional vector space. A second method of constructing a symplectic ma- Although one cannot make proper drawings of such a trix can be quicker and easier. We first construct a space one can certainly work in the space mathemati- Hamiltonian matrix as described above and then take cally and calculate arithmetic means and 10 ×10 vari- its matrix exponential using Matlab. The result is ance-covariance matrices.18-20 (Equation 61 is, in fact, symplectic. See Example 2 in the Appendix. true for Hamiltonian matrices of any size.) In order to test whether or not a particular matrix Augmented symplectic matrices is Hamiltonian one can check whether or not it obeys Equation 46. Or it may be easy to check whether all For heterocentric systems it is sometimes conven- three Equations 58 to 60 are obeyed, that is, whether ient to work with augmented symplectic matrices. the two off-diagonal submatrices are symmetric and They take the form one diagonal submatrix is the negative of the transpose of the other. See the Appendix for examples.  S δ Constructing a Hamiltonian matrix is just as easy. T =   (62)  T  One chooses N and P in Equation 55 arbitrarily except o 1 that they must be symmetric. One can then choose M arbitrarily in which case Q is the negative of the where S is symplectic. Thus an augmented symplec- transpose of M. tic matrix is a symplectic matrix with an additional right-hand column and an additional trivial bottom How to construct a symplectic matrix row. If S is 4× 4 then T is 5× 5 . Submatrix δ is any 4×1 matrix and o is the 4×1 null matrix. The One occasionally wishes to construct a symplec- bottom row of T consists of four 0s and a 1. For eve- tic matrix. As mentioned above this is not difficult if ry augmented symplectic matrix there corresponds an the matrix is 2× 2 : one can usually chose any three optical system.12 of the four entries and then calculate the fourth from From the definition of the determinant the requirement that the matrix has a unit determinant det T = 1× det S . Hence from Equation 20 (Equation 20). The task is much harder if the matrix det T = 1. (63) × is 4 4 . We describe two methods below. T  I ζI T is not an augmented symplectic matrix.   Multiplication according to Equation 2 shows that Consider the matrix   where ζ is a scalar. O I  S−1 −S−1δ Testing shows that it is symplectic. Now consider the ma- T−1 =   (64)  T   I O  o 1  trix   . In general it is not symplectic because   which is an augmented symplectic matrix. C I  The product of two augmented symplectic matri- Equation 25 fails. However if we restrict C to being ces is an augmented symplectic matrix as the follow- symmetric then we see that all of Equations 25 to 27 ing shows: are obeyed and so the matrix is symplectic. Because S S S δ + δ  1 2 1 2 1 . symplectic matrices are closed under multiplication T1T2 =  T  (65) we can construct symplectic matrices by multiplying  o 1  any number of matrices of these two forms. (The first Thus, like symplectic matrices, augmented symplec- matrix is the transference of a homogeneous gap and tic matrices are closed under matrix multiplication. the second that of a thin system. Instead of the first of They are also not closed under addition or under mul- these two matrices one can also use any matrix of the tiplication by a scalar.

9 The South African Optometrist S Afr Optom 2010 69(1) 3-13 WF Harris - Symplecticity and relationships among the fundamental properties in linear optics

An augmented symplectic matrix can be parti- is augmented symplectic. tioned as If Hamiltonian matrix H is 4× 4 then augmented Hamiltonian matrix G is 5× 5 . The bottom row be-  A B e    ing trivial G has 20 nontrivial entries. Submatrix β T =  C D ππ . (66)   adds four degrees of freedom to the 10 of H. Thus G has 14 degrees of freedom. oT oT 1    The augmented Hamiltonian matrices define a 14- dimensional vector space in which one can calculate It follows from Equations 39 and 64 that arithmetic means and 14 ×14 variance-covariance matrices.19  T T T T   D −B − D e + B ππ −1  T T T T  Concluding remarks T = −C A C e − A ππ  . (67)    oT oT 1  The intentions here have been to bring together in   one place basic results in the context of symplecticity Because submatrix S of augmented symplectic which continue to be of use in work in the optics of matrix T is symplectic all of the results for symplectic vision. matrices (symmetric products, Schur complements, From the definition of a symplectic matrix we see etc.) apply in the context of augmented symplectic that the inverse and transpose of a symplectic matrix matrices as well. are also symplectic and that symplectic matrices (of the same size) are closed under matrix multiplication but not under addition or multiplication by a scalar. Augmented Hamiltonian matrices Although the fundamental properties may be symmetric or asymmetric symplecticity makes certain As for symplectic matrices one can define an pairwise products necessarily symmetric. We now augmented Hamiltonian matrix to be a Hamiltonian have the following rule: the product of two properties matrix with an additional right-hand column and an in the same row of a transference is symmetric if the additional bottom row. An augmented Hamiltonian second of the pair is transposed, and the product of matrix takes the form two properties in the same column is symmetric if the T T  H β first is transposed. Thus CD (same row) and C A G =   (68) (same column), for example, are symmetric while  T  T T o 0 D C, AC , DC and AT , for example, may or may not be symmetric. where H is Hamiltonian and β is arbitrary. The bot- The product of a fundamental property and the in- tom row is a row of zeros. verse of another fundamental property is also sym- It is obvious that augmented Hamiltonian matri- metric provided the two properties are in the same ces are closed under addition and multiplication by row with the first inverted or in the same column with _ _ a scalar. Thus the arithmetic average of augmented the second inverted. For example D 1C and CA 1 Hamiltonian matrices is augmented Hamiltonian. (the negatives of the front- and back-vertex powers of _ _ Augmented Hamiltonian matrices bear the same a system23) are symmetric while CD 1 and A 1C are relationship to augmented symplectic matrices as not generally symmetric. Hamiltonian matrices do to symplectic matrices.18 Of course where inverses are involved the equa- That is, if T is augmented symplectic then tions are meaningful only if the matrices in question LogT = G (69) are nonsingular. Furthermore one needs to be aware is augmented Hamiltonian and if G is augmented in numerical work that inversion of nearly-singular Hamiltonian then matrices can lead to spurious results. _1 G B A is the corneal plane refractive compensation e = T (70) for an eye.21 If it were not symmetric it would not be

10 The South African Optometrist S Afr Optom 2010 69(1) 3-13 WF Harris - Symplecticity and relationships among the fundamental properties in linear optics possible properly to compensate for the refractive er- 13. Harris WF. The average eye. Ophthal Physiol Opt 2004 _ ror with thin lenses. But B 1A is symmetric because 24 580-585. of symplecticity and so we have the remarkable fact 14. van Gool RD, Harris WF. The concept of the average eye. Optom Vis Sci 2004 81 (12S) 163. that it is possible to compensate for the refractive er- 15. van Gool RD, Harris WF. The concept of the average eye. ror of an eye whose power ( F = −C ) is asymmetric35 S Afr Optom 2005 64 38-43. by means of thin lenses (the usual spherocylindrical 16. Harris WF. The log-transference and an average Gaussian lenses of optometry) whose powers are symmetric. It eye. S Afr Optom 2005 64 84-88. 17. Harris WF, Cardoso JR. The exponential-mean-log-trans- would seem, therefore, that were it not for symplec- ference as a possible representation of the optical character ticity there might have been no optometry! of an average eye. Ophthal Physiol Opt 2006 26 380-383. An subsequent paper22 uses many of the results giv- 18. Cardoso JR, Harris WF. Transformations of ray transfer- en here including all the Schur complements (Equa- ences of optical systems to augmented Hamiltonian matri- tions 40 to 43) and many of the symmetric products. ces and the problem of the average system. S Afr Optom 2007 66 56-61. In numerical work it is often useful to be able to 19. Harris WF. Quantitative analysis of transformed transfer- recognize and construct symplectic and Hamiltonian ences of optical systems in a space of augmented Hamilto- matrices. This paper shows how. Numerical exam- nian matrices. S Afr Optom 2007 66 62-67. ples are treated in the Appendix. 20. Mathebula SD, Rubin A, Harris WF. Quantitative analysis in Hamiltonian space of the ray transference of a cornea. S Afr Optom 2007 66 68-76. Acknowledgements 21. Harris WF. A unified paraxial approach to astigmatic optics. Optom Vis Sci 1999 76 480-499. I gratefully acknowledge financial support from 22. Harris WF. Special rays and structures in general optical the National Research Foundation. I thank R D van systems. S Afr Optom 2010 69 submitted herewith. Gool, T Evans and A Rubin for discussions and for 23. Harris WF. Back- and front-vertex powers of astigmatic systems. Optom Vis Sci 2010 87 70-72. commenting on the manuscript. 24. Schur I. Über Potenzreihen, die im Innern des Einheitkreis- es beschränkt sind. Reine Angew Math 1917 147 205-232. References 25. Zhang F. Editor of: The Schur Complement and Its Applica- tions. Springer, New York, 2005. 1. Guillemin V, Sternberg S. Symplectic Techniques in Phys- 26. Dopico FM, Johnson CJ. Complementary bases in sym- ics. Cambridge University Press, Cambridge, 1984. plectic matrices and a proof that their determinant is one. 2. Serre D. Matrices: Theory and Applications. Springer-Ver- Linear Algebra Its Appl 2006 419 772-778. lag, New York, 2002. 27. Harris WF. Magnification, blur, and ray state at the retina 3. Kauderer M. Symplectic Matrices: First Order Systems and for the general eye with and without a general optical instru- Special Relativity. World Scientific, Singapore, 1994. ment in front of it. 1. Distant objects. Optom Vis Sci 2001 4. McDuff D, Salamon D. Introduction to Symplectic Topol- 78 888-900. ogy. 2nd ed, Oxford University Press, Oxford, 1998. 28. Harris WF. Nodes and nodal points and lines in eyes and 5. Anton H, Rorres C. Elementary Linear Algebra: Applica- other optical systems. Ophthal Physiol Opt 2010 30 24-42. tions Version. 9th ed, Wiley, Hoboken NJ, 2005. 29. Harris WF. Cardinal points and generalizations. Ophthal 6. Hohn FE. Elementary Linear Algebra. Macmillan, New Physiol Opt 2010 30 submitted. York, 1973. Republished by Dove, Mineola NY, 2002. 30. Higham NJ. Functions of Matrices: Theory and Computa- 7. Meyer CD. Matrix Analysis and Applied Linear Algebra. tion. SIAM, Philadelphia, 2008, 323. SIAM, Philadelphia, 2000. 31. Bernstein DS. Matrix Mathematics: Theory, Facts, and 8. Schneider H, Barker GP. Matrices and Linear Algebra. 2nd Formulas with Application to Linear Systems Theory. Prin- ed, Holt, Rinehart and Winston, New York, 1973. Repub- ceton Univ Press, Princeton, 2005, 85. lished by Dover, Mineola NY, 1989. 32. Horn RA, Johnson CR. Topics in Matrix Analysis. Cam- 9. Graybill FA. Matrices with Applications in Statistics. 2nd bridge University Press, Cambridge, 1991. ed, Wadsworth, Belmont CA, 1983, 183-201. 33. Cardoso JR. Logaritmos de Matrizes: Aspectos Teóricos e 10. Torre A. Linear Ray and Wave Optics in Phase Space. El- Numéricos. Doctoral thesis, Department of Mathematics, sevier, Amsterdam, 2005, 11, 192. University of Coimbra, Coimbra, Portugal, 2003. 11. Sudarshan ECG, Makunda N, Simon R. Realization of first 34. Cardoso JR. An explicit formula for the matrix logarithm. S order optical systems using thin lenses. Optica Acta 1985 Afr Optom 2005 64 80-83. 2 855-872. 35. Harris WF. Dioptric power: its nature and its representation 12. Harris WF. Realizability of systems of given linear optical in three- and four-dimensional space. Optom Vis Sci 1997 character. Optom Vis Sci 2004 81 807-809. 74 349-366.

11 The South African Optometrist S Afr Optom 2010 69(1) 3-13 WF Harris - Symplecticity and relationships among the fundamental properties in linear optics

Appendix Consider the 2× 2 matrices first. All we have to do is check the determinant and the trace: if the deter- We illustrate here recognition (Example 1) and minant is 1 then the matrix is symplectic; if the trace construction (Example 2) of symplectic and Hamilto- is zero then the matrix is Hamiltonian. We see that nian matrices. Example 3 compares the logarithm of (a) is both symplectic and Hamiltonian. (b) is Ham- the entries of a particular symplectic matrix with the iltonian and, because its determinant is not 1, it is not matrix logarithm of the matrix. symplectic. (c) is neither symplectic nor Hamiltoni- an. Thus (a) could be the transference of an optical Example 1 Classify each of the following as sym- system but (b) and (c) could not. (a) and (b) could plectic, Hamiltonian or both: both be the log-transferences of optical systems. Consider now the 4× 4 matrices. We check first  1 2  for Hamiltonicity. Partitioning according to Equation   (a) 55 we observe that submatrix P is not symmetric in −1 −1   (d). Hence (d) is not Hamiltonian. N and P are symmetric in (e) and Q is the negative of the transpose of  1 0  M; hence (e) is Hamiltonian. Q is not the negative of (b)     the transpose of M in (f) and so (f) is not Hamiltonian. −1 −1 Checking for symplecticity is not as easy. We resort to substituting into the left-hand side of Equation 17  1 2 and multiplying. For (d) we finally obtain (c)     −1 0

 4 0 0 6    −1 3 6 1  (d)    2 −1 −4 1      which is not E. Hence (d) is not symplectic. We also 0 2 0 −3   do not obtain E for either (e) or (f). Thus none of the matrices is symplectic. Thus none of (d), (e) and (f)  4 0 0 6  could be the transference of an optical system but (e)   could be the log-transference of one. −1 3 6 1    (e) Example 2 Starting with each of the Hamiltonian ma-  2 −1 −4 1    trices in Exercise 1 construct a symplectic matrix.   −1 2 0 −3 (a), (b) and (e) are the only Hamiltonian matrices in Exercise 1. Using Matlab we obtain the matrix ex- ponentials. They turn out to be  4 0 0 6    −1 3 6 1    (f)  2 −1 −4 0      −1 2 1 −3

12 The South African Optometrist S Afr Optom 2010 69(1) 3-13 WF Harris - Symplecticity and relationships among the fundamental properties in linear optics and

approximately.

Example 3 Compare the logarithm applied separately 1 2 to the entries of the symplectic matrix   with   1 3 the principal matrix logarithm of the matrix.

They are and respectively.

13 The South African Optometrist