Symplecticity and relationships among the fundamental properties in linear optics

Symplecticity is of profound significance to modern science and to optometry in particular. It can even be argued (see towards the end of this paper) that it is the reason why refractive errors can be compensated by means of conventional spherocylindrical lenses and is, therefore, no less than a sine qua non of optometry. Symplecticity implies particular relationships among the fundamental optical properties of an optical system. Although there are several good sources that deal with symplecticity1-4 they tend to be mathematically sophisticated and not readily accessible for most people working in visual optics. Because the relationships can take many unfamiliar forms, are not easy to remember, and are often needed in analyses of optical problems it would seem useful to have a compact and more accessible summary. Accordingly the objective of this paper is to supply such a summary.


Introduction
Symplecticity is of profound significance to modern science and to optometry in particular.It can even be argued (see towards the end of this paper) that it is the reason why refractive errors can be compensated by means of conventional spherocylindrical lenses and is, therefore, no less than a sine qua non of optometry.
Symplecticity implies particular relationships among the fundamental optical properties of an optical system.Although there are several good sources that deal with symplecticity 1-4 they tend to be mathematically sophisticated and not readily accessible for most people working in visual optics.Because the relationships can take many unfamiliar forms, are not easy to remember, and are often needed in analyses of optical problems it would seem useful to have a compact and more accessible summary.Accordingly the objective of this paper is to supply such a summary.

Basic results of linear algebra
We make use of the basic results of linear algebra as presented in introductory texts [5][6][7][8] .Our matrices are all real, that is, their entries are all real numbers.In particular for square matrices A and B (AB T ) = B T A T (1) where T A is the matrix transpose of A. Also if = = AB BA I , (2)

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

WF Harris -Symplecticity and relationships among the fundamental properties in linear optics
The South African Optometrist 4 where I is an identity matrix, then B is the inverse of A and is written (3) provided the inverses exist.Also (A _ 1 ) T = (A T ) _ 1 .
(4) We use the common abbreviation A _ T for either side of Equation 4. Also det(AB)=det A det B (5) and det A T =det A .
(6) Partitioned matrices feature importantly in symplecticity.They take the form a matrix of matrices as it were.They are no different from ordinary matrices, however, and obey the usual rules of matrix algebra.Equation 7does not In optical applications S represents the transference of an optical system and the submatrices the ( 2 2 × ) fundamental linear optical properties of the system.
Multiplication takes the form as might be expected.Multiplication by a scalar is as expected: The transposition operator is taken inside the partitioned matrix but the off-diagonal submatrices are also interchanged: It might be supposed that similar simple expressions can be written for the inverse and determinant of a partitioned matrix; that, however, is not the case in general.(There are expressions but they are much more complicated 9 .)

The symplectic unit matrix
Consider the partitioned matrix .It is sometimes called the symplectic unit matrix 10 .Although symbols vary, more often than not it is represented by J. (We use E instead because J is already used for one of the basic matrices in the set I, J, K and L.) Applying Equation 8we find that  (14) It follows from Equation 10that (15) From the definition 5-8 of the determinant it turns out that

Symplectic matrices
By definition a matrix S is symplectic if The transference of an optical system is symplectic 1, 4 .Different optical systems can have the same transference.For every 2 2 × or 4 4 × symplectic matrix it is possible to have an optical system whose transference is that matrix. 11,12 ppose 1 S and 2 S are both symplectic (and have the same n).Consider the product 2 1 S S . Substituting it into the left-hand side of Equation 17 we obtain ( ) ( ) In other words S 1 S 2 satisfies Equation 17 and so is 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 symplectic matrices is symplectic or, in other words, symplectic matrices are closed under multiplication.
On the other hand symplectic matrices are not in general closed under addition or multiplication by a scalar.(The reader is encouraged to show this by working out some simple examples.)This means, in particular, that an arithmetic average of symplectic matrices is not in general symplectic, a fact that has important implications for basic statistics.If one wants to calculate an average eye, for example, one would want the average to be a possible eye.The fact that the arithmetic average of transferences is not symplectic in general implies that, strictly speaking, it is not meaningful to calculate an average eye that way.The problem of how to calculate an average eye is not a simple one.The relationship between symplectic and Hamiltonian matrices, to be discussed below, appears to offer a solution [13][14][15][16][17][18][19][20] .Despite what we have said here the arithmetic average may, in some cases, be sufficiently close to being symplectic for it to be a good enough approximation.
It is easy to see that I and E are themselves symplectic: they satisfy Equation 17.On the other hand O does not and, therefore, is not symplectic.
Making use of Equations 6, 16 and 17 we find that ( ) This suggests that In fact it turns out that 1 det = S .
(20) (The proof is not simple and will not be attempted here.Proofs are given elsewhere 2-4 .)This means a symplectic matrix S is never singular (that is, which shows that T S is symplectic.Let symplectic matrix S be partitioned as in Equation 7.Then, from Equation 10, Substitution into Equation 17 results in Equating the four blocks on the left and right we see that 27) and Transposition applied to Equation 28shows that it is equivalent to Equation 27.When S is 2 2 × Equations 25 and 26 are trivially true and Equation 28 reduces to Equation 20.If the four entries of a 2 2 × matrix are chosen arbitrarily then the matrix is usually not symplectic.To construct a 2 2 × symplectic matrix one is free to choose at most three of the entries arbitrarily; the fourth is determined by Equation 20.One can say that symplecticity implies a loss of one degree of freedom from four to three.Note that Equation 20 implies that a 2 2 × symplectic matrix cannot have a row or a column of zeros.
When S is 4 4 × Equations 25 and 26 imply a loss of one degree of freedom each and Equation 27a loss of four degrees of freedom.Thus, instead of 16 degrees of freedom, a 4 4 × symplectic matrix has only 10 degrees of freedom.Constructing a 4 4 × symplectic matrix, however, is not easy.A couple of methods will be described later.
Because T S is symplectic Equation 22shows that Hence and Equation 33 is equivalent to Equation 32.
In order to test whether a given matrix is symplectic or not one can proceed as follows.One can directly test whether it obeys Equation 17. Alternatively one can test whether all three of Equations 25 to 27 (or Equations 30 to 32) are obeyed.In either case if the matrix does obey the equation or equations then it is symplectic; if it does not then it is not symplectic.Numerical examples are presented in the Appendix.

Symmetric products
Applying Equation 1 to the right-hand side of Equation 25 we obtain In other words A T C is symmetric.The same approach can be applied to Equations 26, 30 and 31.One reaches the conclusion that, although A, B, C and D in S (Equation 7) can be symmetric or asymmetric, all of the following products are necessarily symmetric: Notice that, in these cases, the two submatrices in a product occupy the same row or the same column of S. Furthermore, if the two submatrices share the same column of S then the first of the two is transposed; if the two are in the same row then the second is transposed.As an aid to memory one might say 'rows, transpose second; columns, transpose first'.Other products, including AC, BD provided A is nonsingular.Hence showing that CA _ 1 is symmetric.In the same way one finds other products that are symmetric.Thus

The inverted symplectic matrix
Because of Equations 25 to 28 multiplication shows that for a symplectic matrix S partitioned as in Equation 7. Similarly, because of Equations 30 to 32, one finds that By Equation 2then we see that for any symplectic matrix S. Thus the inverse of a symplectic matrix is what one might expect for a 2 2 × matrix except that the submatrices are also transposed.If S is symplectic then the Schur complements reduce to particularly neat expressions:

The Schur complements
These results were apparently first obtained by Dopico and Johnson 26 .In connection with eyes they have been involved in several papers 22,[27][28][29] although not always recognised as such.
To prove Equation 40 we premultiply each side of Equation 28 by D _ T to give and apply Equation 1 to give

Hamiltonian matrices
There is another class of matrices which is important in modern science and which bears a surprising relationship to symplectic matrices: it is the class of Hamiltonian matrices.A matrix H is Hamiltonian if it obeys (linear optics).It is a remarkable fact that the principal matrix logarithm of a symplectic matrix is a Hamiltonian matrix and the matrix exponential of a Hamiltonian matrix is symplectic 10, 17, 32-34 .The exponential of a real square matrix X is the real infinite convergent series Any matrix X which satisfies is called a logarithm of A; in general there is more than one.However, if none of the eigenvalues of A is zero or a negative real number, then A does have a unique real logarithm the magnitude of whose eigenvalues are less than π .It is the principal logarithm and is written A Log .So, if S is symplectic then is Hamiltonian and if H is Hamiltonian then is symplectic.
It is important to note that these are not the familiar logarithm and exponential simply applied to the entries of the matrix separately.(In Matlab they are given by the functions logm and expm as opposed to log and exp which operate separately on the entries of the matrix.)See Example 3 in the Appendix. Let Addition leads to ( ) ( ) and so ( ) ( ) which shows that It is apparent from Equation 46that O and E are Hamiltonian but that I is not.Suppose H is partitioned as If H is Hamiltonian then according to Equations 46 and 11 This shows that In other words, for a Hamiltonian matrix partitioned as in Equation 55, the off-diagonal submatrices N and P are necessarily symmetric and one diagonal submatrix is the negative of the transpose of the other.For a 2 2 × Hamiltonian matrix Equations 58 and 59 are trivially true and Equation 60 is equivalent to the statement that tr 0 t r = H (61) or, in other words, that a Hamiltonian matrix has zero trace or is traceless as it is sometimes called.Thus, as in the case of symplectic matrices, there is a loss of one degree of freedom from four to three.In fact the 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 space.Individual matrices can be plotted in the space to form trajectories and clusters.An arithmetic mean is a point in the same space at the centre of a cluster.
3 3× variance-covariance matrices provide measures of spread and variation in the space.

WF Harris -Symplecticity and relationships among the fundamental properties in linear optics
The South African Optometrist 9 59 each represent a loss of a degree of freedom and Equation 60 a loss of four degrees of freedom.Thus, also as for symplectic matrices, there is a loss of six degrees of freedom from 16 to 10.The 4 4 × Hamiltonian matrices define a 10-dimensional vector space.9][20] (Equation 61 is, in fact, true for Hamiltonian matrices of any size.) In order to test whether or not a particular matrix is Hamiltonian one can check whether or not it obeys Equation 46.Or it may be easy to check whether all three Equations 58 to 60 are obeyed, that is, whether the two off-diagonal submatrices are symmetric and one diagonal submatrix is the negative of the transpose of the other.See the Appendix for examples.
Constructing a Hamiltonian matrix is just as easy.One chooses N and P in Equation 55 arbitrarily except that they must be symmetric.One can then choose M arbitrarily in which case Q is the negative of the transpose of M.

How to construct a symplectic matrix
One occasionally wishes to construct a symplectic matrix.As mentioned above this is not difficult if the matrix is 2 2 × : one can usually chose any three of the four entries and then calculate the fourth from the requirement that the matrix has a unit determinant (Equation 20).The task is much harder if the matrix is 4 4 × .We describe two methods below.where B is symmetric but an optical system with this transference is not simple.) A second method of constructing a symplectic matrix can be quicker and easier.We first construct a Hamiltonian matrix as described above and then take its matrix exponential using Matlab.The result is symplectic.See Example 2 in the Appendix.

Augmented symplectic matrices
For heterocentric systems it is sometimes convenient to work with augmented symplectic matrices.They take the form where S is symplectic.Thus an augmented symplectic matrix is a symplectic matrix with an additional right-hand column and an additional trivial bottom row.If S is 4 4 × then T is 5 5 × .Submatrix δ is any 1 4 × matrix and o is the 1 4 × null matrix.The bottom row of T consists of four 0s and a 1.For every augmented symplectic matrix there corresponds an optical system. 12rom the definition of the determinant S T det 1 det × = . Hence from Equation 20 T T is not an augmented symplectic matrix.Multiplication according to Equation 2 shows that which is an augmented symplectic matrix.
The product of two augmented symplectic matrices is an augmented symplectic matrix as the following shows: Thus, like symplectic matrices, augmented symplectic matrices are closed under matrix multiplication.They are also not closed under addition or under multiplication by a scalar.Because submatrix S of augmented symplectic matrix T is symplectic all of the results for symplectic matrices (symmetric products, Schur complements, etc.) apply in the context of augmented symplectic matrices as well.

Augmented Hamiltonian matrices
As for symplectic matrices one can define an augmented Hamiltonian matrix to be a Hamiltonian matrix with an additional right-hand column and an additional bottom row.An augmented Hamiltonian matrix takes the form where H is Hamiltonian and β is arbitrary.The bottom row is a row of zeros.
It is obvious that augmented Hamiltonian matrices are closed under addition and multiplication by a scalar.Thus the arithmetic average of augmented Hamiltonian matrices is augmented Hamiltonian.
Augmented Hamiltonian matrices bear the same relationship to augmented symplectic matrices as Hamiltonian matrices do to symplectic matrices. 18hat is, if T is augmented symplectic then The augmented Hamiltonian matrices define a 14dimensional vector space in which one can calculate arithmetic means and 14 1 4 1 4 ×14 variance-covariance matrices. 19

Concluding remarks
The intentions here have been to bring together in one place basic results in the context of symplecticity which continue to be of use in work in the optics of vision.
From the definition of a symplectic matrix we see that the inverse and transpose of a symplectic matrix 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.
Although the fundamental properties may be symmetric or asymmetric symplecticity makes certain pairwise products necessarily symmetric.We now have the following rule: the product of two properties in the same row of a transference is symmetric if the second of the pair is transposed, and the product of two properties in the same column is symmetric if the first is transposed.Thus CD T (same row) and C T A (same column), for example, are symmetric while D T C, AC T , DC and AT , for example, may or may not be symmetric.
The product of a fundamental property and the inverse of another fundamental property is also symmetric provided the two properties are in the same row with the first inverted or in the same column with the second inverted.of symplecticity and so we have the remarkable fact that it is possible to compensate for the refractive error of an eye whose power ( C F − = ) is asymmetric 35 by means of thin lenses (the usual spherocylindrical lenses of optometry) whose powers are symmetric.It would seem, therefore, that were it not for symplecticity there might have been no optometry!An subsequent paper 22 uses many of the results given here including all the Schur complements (Equations 40 to 43) and many of the symmetric products.
In numerical work it is often useful to be able to recognize and construct symplectic and Hamiltonian matrices.This paper shows how.Numerical examples are treated in the Appendix.

Appendix
We illustrate here recognition (Example 1) and construction (Example 2) of symplectic and Hamiltonian matrices.Example 3 compares the logarithm of the entries of a particular symplectic matrix with the matrix logarithm of the matrix.
Example 1 Classify each of the following as symplectic, Hamiltonian or both: Consider the 2 2 × matrices first.All we have to do is check the determinant and the trace: if the determinant is 1 then the matrix is symplectic; if the trace is zero then the matrix is Hamiltonian.We see that (a) is both symplectic and Hamiltonian.(b) is Hamiltonian and, because its determinant is not 1, it is not symplectic.(c) is neither symplectic nor Hamiltonian.Thus (a) could be the transference of an optical system but (b) and (c) could not.(a) and (b) could both be the log-transferences of optical systems.
Consider now the 4 4 × matrices.We check first for Hamiltonicity.Partitioning according to Equation 55we observe that submatrix P is not symmetric in (d).Hence (d) is not Hamiltonian.N and P are symmetric in (e) and Q is the negative of the transpose of M; hence (e) is Hamiltonian.Q is not the negative of the transpose of M in (f) and so (f) is not Hamiltonian.Checking for symplecticity is not as easy.We resort to substituting into the left-hand side of Equation 17and multiplying.For (d) we finally obtain which is not E. Hence (d) is not symplectic.We also do not obtain E for either (e) or (f).Thus none of the matrices is symplectic.Thus none of (d), (e) and (f) could be the transference of an optical system but (e) could be the log-transference of one.
both sides of Equation 17 and premultiplying by S and postmultiplying by 1 − etc., are not symmetric in general.Postmultiplying Equation 25 by 1 −A and premul-

For
Schur complement of A in S. Similarly there are Schur complements of B, C and D in S. Schur complements arose out of the work of I Schur 24 and are of considerable modern scientific interest 25 .It is no surprise that they should arise in visual optics.
43 follow similarly.Thus the Schur complements of the block-diagonal submatrices of transference S are the transposed inverses of their opposites while the Schur complements of the other submatrices are the negatives of the transposed inverses of their opposites.To recall the sequence of submatrices on the left-hand sides of Equations 40 to 43 the following may be helpful: beone writes the submatrices in cyclical order going clockwise for the block-diagonal matrices and anticlockwise for the others.
Hamiltonian matrices are closed under addition.That Hamiltonian matrices are closed under addition and multiplication by a scalar means that the arithmetic average of Hamiltonian matrices is itself Hamiltonian.
it is not symplectic because Equation 25 fails.However if we restrict C to being symmetric then we see that all of Equations 25 to 27 are obeyed and so the matrix is symplectic.Because symplectic matrices are closed under multiplication we can construct symplectic matrices by multiplying any number of matrices of these two forms.(The first matrix is the transference of a homogeneous gap and the second that of a thin system.Instead of the first of these two matrices one can also use any matrix of the form The bottom row being trivial G has 20 nontrivial entries.Submatrix β adds four degrees of freedom to the 10 of H. Thus G has 14 degrees of freedom. of the front-and back-vertex powers of a system 23 ) are symmetric while CD _ 1 and A _ 1 C are not generally symmetric.Of course where inverses are involved the equations are meaningful only if the matrices in question are nonsingular.Furthermore one needs to be aware in numerical work that inversion of nearly-singular matrices can lead to spurious results.B _ 1 A is the corneal plane refractive compensation for an eye. 21If it were not symmetric it would not be π compensate for the refractive error with thin lenses.But B _ 1 A is symmetric because

Example 2
Starting with each of the Hamiltonian matrices in Exercise 1 construct a symplectic matrix.(a), (b) and (e) are the only Hamiltonian matrices in Exercise 1.Using Matlab we obtain the matrix exponentials.They turn out to be )