Appendix A Diagonalization of an S-matrix Consider the standard basis B2 = (el' e2) in a two-dimensional vector space over the complex numbers V2. Denote in V2 a complex symmetric invertible S-matrix S as (A.1) For any matrix M in V2, let Mt and Mt denote respectively the transpose and the adjoint of M. We will prove Theorem. Let S be as given in (A. i). Let f.L! and f.L§ be the eigenvalues of the positive-definite matrix st S. Then, there exists in V2 (i) a unitary matrix A such that the the "unitary congruent transforma­ tion" AtSA reduces S to a real diagonal matrix § as (A.2) _ (ii) a unitary matrix B such that Bt SB reduces to a real diagonal matrix S' as We observe that the unitary congruent transformation on complex sym­ metric matrices does what the unitary (similarity) transformation does on normal matrices. Proof. There are two cases. 331 332 Diagonalization of an S-matrix For the positive-definite matrix st S, there exists a unitary matrix U such that Ut(stS)U = (~I :~), J1~ =1= J1~, (A.4) where Ut is the adjoint of U. Since S is complex symmetric and U is unitary, the left side of (A.4) can be decomposed as (A.5) where "*,, stands for complex conjugation. Let us denote w= UtSU. (A.6) W is a "unitary congruent transform" of S. It is complex symmetric and invertible because S is. In terms of W, (A.4) is expressed as w*w = (J1I (A.7) o J1~0) which shows that W is hermitian-orthogonal in the principal-axis basis whose basis elements in V2 are the normalized eigenvectors of st S. Since W is complex symmetric, so is W t . Therefore, it follows that Whence wtw=wwt, (A.9) showing that the unitary congruent transform W of S is normal. Let w = (a ac _ b2 =1= O. (A. 10) be'b) Since W is nonsingular and W t = W*, we get from (A.8) and (A.9) 2 w- 1 (c*J11 -b*J12)1 (A.ll) - qet W* -b* J1~ a* J1~ . Diagonalization of an S-matrix 333 Since W is complex symmetric and J..L~ =1= J..L~, it follows from (A.ll) that b* = 0 ==> b = O. (A.12) Therefore, Win (A.lO) takes the form W = (~ ~), ac =1= O. (A.13) Thus (A.7) can be expressed as (A.14) Whence" (A.15) where a and (3 are arbitrary in [0,211"). We now write (A.13) as ia W = Ut SU = (J..Ll (e (A.16) o J..L20) 0 e~(3~). Let A=UF, (A.17) where U is as given by (A A) and F is a unitary matrix of the form e-ia/2 0 ) F = ( 0 e-i(3/2 . (A.18) We form a new unitary congruent transform W' as (A.19) where we used the fact that F is complex symmetric. Since both Wand F are diagonal, they commute with each other. Therefore, (A.19) yields a real diagonal matrix as W' = FW F = W F2 = (~l :2)' (A.20) which is (A.2). Case (i) is proved. 334 Diagonalization of an S-matrix Case (ii) : f-LI = f-L~ == f-L2 We assume without essential loss of generality that Sl =F 0, S2 =F 0, and S2 is real. Under these conditions, it it easy to show that f-LI = f-L~ == f-L2 leads to (A.21) (A.22) and (A.23) where I denotes the identity matrix in V2. Let T = ~ S = (~ ~* ), (A.24) where a = sI/f-L, 13 = S2/ f-L, 13 is real. Now, it is easily seen that T is normal. In fact, it is unitary and complex symmetric. We need the following Lemma. For a matrix T as given by (A.24), there exists an orthogonal and involutory matrix V with det V = -1 such that VtTV = V-lTV = VTV = (~ _~* ) (A.25) where V t and V-I denote respectively the transpose and the inverse of V, and v is a complex number. Proof. Let Tx= AX, (A.26) where A is an eigenvalue of T. We find that Al = t[(a-a*) +d], } (A.27) A2 = "2 [(a - a*) - d], d = v(a + a*)2 + 4132 > ° (A.28) Diagonalization of an S-matrix 335 and Xl = ~ ( f ), for A!, (A.29) X2 = ~ ( ~p ), for A2, (A.30) p= 2~ [(a+a*)+d], (A.31) N= V1+p2. (A.32) We note that the eigenvectors Xi are real-valued. The desired unitary matrix V is 1 ) (A.33) V=-.!..(pN 1 -p , which is orthogonal and involutory with det V = -1, as is easily seen. From (A.24) and (A.33) we get VtTV = V-lTV = VTV = 1 ( ap2 + 2{3p - a* p(a + a*) + {3(1 - p2)) (A 34) N2 p(a + a*) + {3(1- p2) -(a*p2 + 2{3p - a) . It is easy to show that p(a + a*) + {3(1 - p2) = 0, so that with 1 v = N2 [ap2 + 2{3p - a* ], (A.35) the lemma is proved. Since S = p,T from (A.24), (A.25) can be writen as t 0 ) V SV = P, (v0 -v* , (A.36) which is a unitary congruent transform of S. Now, (A.37) 336 Diagonalization of an S-matrix Also, by the fact that V is orthogonal and S is complex symmetric, we can write V-l(stS)V = (VtSV)*(VtSV) = J.£ 2 1111 2 I. (A.38) Comparison of (A.37) and (A.38) yields (A.39) whence (AAO) where'IjJ is arbitrary in [0, 27r). (A.36) is expressed as ei'/fr 0 ) V t SV = J.£ ( 0 _e-i'/fr . (AA1) Let B=VG, (AA2) where G is a unitary matrix of the form e-i'/fr/2 0 ) G = ( 0 ei'/fr/2 . (AA3) Evidently, G is complex symmetric also. Since the diagonal matrices G and V t SV commute, it follows that BtSB _ Gt(vtSV)G = G(VtSV)G - (VtSV)G2 = J.£ (~ ~1)' (AA4) which is (A.3). The proof of the theorem is complete. Appendix B A Deficient System of Equations A large number of physical problems involving integral equations, differential equations, the Helmholtz integral representation, the Franz integral repre­ sentations, the Cauchy integral, the Plemelj formulas, linear regression, etc., are all reducible to solving deficient linear systems of equations of the form Ax=b (B.I) where A is an n x m complex rectangular matrix, b is a given n-dimensional complex vector, and x is an m-dimensional complex vector which we seek. In fact, every linear transformation can be represented by a matrix. x resides in the domain space Vm (or the row-vector space) of A and b in the range space Vn (or the column-vector space) of A . (B.I) always has a solution in the sense which will be explained later, and can be solved by numerical tech­ niques such as the singular value decomposition or the gradient ptojection method (cf. A. Albert [1] or K. Atkinson [2]). C. Lanczos [3], on whose work this appendix is based, guides us to see the behind-the-scene story of the pseudoinversion by using a single mathematical notion as an essential tool that the solution of an equation such as (B.I) depends fundamentally on the solutions of the related eigenequations and the number of nonzero eigenvalues of the related eigenoperators determines the nature of the solution of the equation. This appendix is a short summary on the general deficient systems of equations. It should be pointed out that both Lanczos' exposition and the Moore-Penrose theory of pseudoinverse are essentially different ways of stating the well-known algebraic relation that the rank of A + the nullity of A = m in the mapping A : Vm ( row-vector space) -+ Vn ( column-vector space). The essential features of Lanczos' argument can be exhibited in a spe­ cial case of the n x n hermitian matrix and will be discussed first and the knowledge gained there extended to the general case of the n x m rectangular matrix. 337 338 A Deficient System of Equations Case of an Hermitian Matrix Let S be an n x n hermitian matrix with reference to a prescribed basis in an n-dimensional complex vector space Vn . We wish to solve a linear system of equations Sx=h. (B.2) We know from the hermiticity of S that there exists a principal-axis basis in Vn , with reference to which (B.2) becomes uncoupled as Dx' = h', (B.3) where o D= (BA) o The diagonal entries Ai are the eigenvalues of S, and the new vectors x' and h' are related to the old ones by x' (B.5) h' (B.6) where T is an n-dimensional unitary matrix whose n columns are made of the normalized eigenvectors belonging to the eigenvalues Ai of S, and Tt is the adjoint of T. Since TtT = TTt = In, where In is the n x n identity matrix, (B.5) and (B.6) may be expressed as x = Tx', h = Tb'. On substituting these results into (B.2), we obtain STx' = Th'. Whence TtSTx' = h'. Comparing the last result with (B.3), we get D = Tt ST. Whence (B.7) We have decomposed in (B.7) the hermitian matrix S in terms of the unitary matrix T and the diagonal matrix D. While this decomposition per A Deficient System of Equations 339 se does not solve (B.2), it will help us to see what can go wrong in solving (B.2) and thereby suggest a method for obtaining a best possible solution for it.