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2 Hermitian kernels

Matrix theory ∗ s×s First consider a Hermitian matrix N = N ∈ C . Its eigenvalues ν1, · · · ,νs are the roots of det(N − νI) = 0. They are real. Corresponding to νj is an eigenvector pj satisfying Npj = νjpj. ∗ These are orthonormal: pj pk = δjk where δjj = 1 and δjk = 0 for j 6= k. Set P = (p1, · · · ,ps). If N and its eigenvalues are real, then P can be taken to be real. The spectral decomposition of N in terms of its eigenvalues and eigenvectors is

s ∗ ∗ N = P ΛP = νjpjpj where Λ = diag(ν1, · · · ,νs), (2.1) Xj=1 s ∗ ∗ ∗ ∗ νjpjpj = P P = Is = P P = (pj pk). Xj=1 So for α ∈ C s α α ∗ α ∗ N = P Λ P = νj pjpj , (2.2) Xj=1 provided that if det(N) = 0, then α has non-negative real part. So for det(N) 6= 0,

s −1 −1 ∗ Nf = g ⇒ f = N g = νj pj(pj g). Xj=1 Similarly for ν not an eigenvalue and f, g ∈ Cs,

s −1 −1 ∗ (νI − N)f = g ⇒ f = (νI − N) g = (ν − νj) pj(pj g). Xj=1 For large n, if s r1 = |ν1| = · · · = |νM | > r0 = max |νj| (2.3) j=M+1 then

M n n n n ∗ N = r1 Cn + O(r0 ) where Cn = [sign (νj)] pjpj = O(1) (2.4) Xj=1 assuming that s does not depend on n. Function theory Now consider a function N(y, z):Ω2 → Cs×s and a σ-finite measure µ on Ω ⊂ Rp. Its integral operator with respect to µ is N defined by (1.1). Suppose that the kernel N is Hermitian, that is, ∗ N(y, z) = N(z,y).

2 2 Then the analogues of the matrix results above are as follows. Suppose that ||N ||2 > 0 and

s |Njj(x,x)dµ(x) < ∞. Z Xj=1

The spectral decomposition of N in terms of its eigenvalues and vector eigenfunctions {pj(y)} :Ω → Cs, is

∞ ∗ ∗ N(y, z) = P (y)ΛP (z) = νjpj(y)pj(z) (2.5) Xj=1

where Λ = diag(ν1,ν2, · · · ), P (y) = (p1(y),p2(y), · · · ).

The eigenfunctions are orthonormal with respect to µ:

∗ p pkdµ = δjk. Z j

−1 In Fredholm theory the convention is to call {λj = νj } the eigenvalues, rather than {νj}. ∗ Iyz = P (y)P (z) is generally divergent but can be thought of as a generalized Dirac function: ∗ ∗ ∗ Iyzf(z)dµ(z)= f(y). P (z) P (y) = (pj(z) pk(y)) satisfies P (z) P (z)dµ(z)= I∞. R n−1 R For n = 1, 2, · · · ,Nn(y, z)= N N(y, z) satisfies

s n ∗ n ∗ Nn(y, z)= P (y)Λ P (z) = νj pj(y)pj(z) . Xj=1

For large n, if (2.3) holds then

M n n n ∗ Nn(y, z)= r1 Cn(y, z)+ O(r0 ) where Cn(y, z)= [sign (νj)] pj(y)pj(z) = O(1). Xj=1

Conditions for (2.5) to hold pointwise and uniformly are given in Withers (1974, 1975, 1978.) It is known as Mercer’s Theorem. The resolvent Given functions f, g :Ω → Cs, the Fredholm integral equation of the second kind,

p(y) − λN p(y)= f(y), (2.6) can be solved for λ not an eigenvalue using

−1 −1 (I − λN ) = I + λNλ, that is, Nλ = (I − λN ) N where

Nλf(y)= Nλ(y, z)f(z)dµ(z), g(z)Nλ = g(y)Nλ(y, z)dµ(y), Z Z and the resolvent of N ,

−1 2 s×s Nλ(y, z) = (I − λN ) N(y, z) : C× Ω →C

3 with operator Nλ is the unique solution of

(I − λN )Nλ = N = Nλ(I − λN ), that is, λN Nλ(y, z)= N(y, z) − Nλ(y, z)= λNλ(y, z)N . If this can be solved analytically or numerically, then one has a solution of (2.6) without the need to compute the eigenvalues and eigenfunctions of N . The resolvent satisfies ∞ ∗ Nλ(y, z)= pj(y)pj(z) /(λj − λ). (2.7) Xj=1

Conditions for this to hold are given by Corollary 3 of Withers (1975). The Fredholm equation of the second kind, (2.6), has solution

∞ ∗ p(y)= f(y)+ λ pj(y) p fdµ/(λj − λ). Z j Xj=1

The resolvent exists except for λ = λj, an eigenvalue. The eigenvalues of N are the zeros of its Fredholm determinant

λ ∞ D(λ) = Πj=1(1 − λ/λj) = exp{− dλ trace Nλ(x,x)dµ(x)}. (2.8) Z0 Z The Fredholm integral equation of the first kind

λN p(x)= p(x) has a solution provided that λ is an eigenvalue. For ν an eigenvalue, its general solution p(x) is a linear combination of the eigenfunctions {pj(x)} corresponding to λj = λ.

I r Example 2.1 Suppose that Y,Z ∈ R and Y ∼ N (0,V ) where V = . So V is the Z 2 r I Y
1/2 correlation matrix for Z . For j ∈ N+, x ∈ R set pj(x) = Hj(x)/j! where Hj(x) is the standard univariate Hermite
polynomial. Then pjpkφI = δjk and

∞ R j r pj(y)pj(z)= φC (y, z)/φI (y)φI (z). Xj=0

This is Mehler’s expansion for the standard bivariate normal distribution. Pearson gave an in- k tegrated version and Kibble extended it to an expansion for φV (x)/φI (x) for x ∈ R and V a correlation matrix. See for example (45.52) p127 and p321-2 of Kotz, Balakrishnan and Johnson (2000).

4 3 Functions of two variables with diagonal Jordan form

Diagonal Jordan form for matrices s×s Consider N ∈ C with eigenvalues ν1, · · · ,νs, the roots of det(N − νI) = 0.N is said to have diagonal Jordan form (DJF) if

∗ ∗ N = P ΛQ where PQ = I and Λ = diag(ν1, · · · ,νs).

So s ∗ ∗ N = νjpjqj , qj pk = δjk where P = (p1, · · · ,ps), Q = (q1, · · · ,qs), Xj=1 s ∗ ∗ ∗ ∗ pjqj = PQ = Is = Q P = (qj pk). Xj=1

If N, Λ are real then P,Q can be taken as real. Also

∗ ∗ NP = P Λ, Npj = νjpj,N Q = QΛ,N qj = νjqj, (3.1) and for any complex α, s α α ∗ α ∗ N = P Λ Q = νj pjqj Xj=1 provided that if det(N) = 0, then α has non-negative real part. Suppose that

iθj s νj = rje and r1 = · · · = rR > r0 = max rj. (3.2) j=R+1

Then

R n n n inθj ∗ N = r1 Cn + O(r0 ) where Cn = e pjqj = O(1) (3.3) Xj=1

Taking α = −1 gives the inverse of N when this exists. Diagonal Jordan form for functions Now consider a function N(y, z):Ω2 → Cs×s. When N has diagonal Jordan form (for example when its eigenvalues are all different), then the Fredholm equations of the first kind,

∗ λN p(y)= p(y), λ¯N q(z)= q(z), or equivalently for ν = λ−1, ∗ N p(y)= νp(y), N q(z) =νq ¯ (z), −1 also have only a countable number of solutions, say {λj = νj ,pj(y),qj(z), j ≥ 1} up to arbitrary constant multipliers for {pj(y), j ≥ 1}, satisfying the bi-orthogonal conditions

∗ q pkdµ = δjk. Z j

5 These are called the eigenvalues and right and left eigenfunctions of (N,µ) or N . Also

∞ ∗ ∗ N(y, z) = νjpj(y)qj(z) = P (y)ΛQ(z) (3.4) Xj=1

where Λ = diag(ν1,ν2, · · · ), P (y) = (p1(y),p2(y), · · · ), Q(z) = (q1(y),q2(y), · · · ). with convergence in L2(µ × µ), or pointwise and uniform under stronger conditions. If N is a real function and Λ is real, then P,Q can be taken as real functions For n ≥ 1,

n−1 Nn(y, z)= N N(y, z) (3.5) satisfies ∞ n ∗ n ∗ Nn(y, z) = νj pj(y)qj(z) = P (y)Λ Q(z) , (3.6) Xj=1 ∞ n n ∗ N p(y) = ν pj(y) q pdµ. j Z j Xj=1

If (3.2) holds then

R n n inθj ∗ Nn(y, z) = r1 Cn + O(r0 ) where Cn = e pj(y)qj(z) = O(1) (3.7) Xj=1 R n n n inθj ∗ N p(y) = r cn + O(r ) where cn = e pj(y) q pdµ = O(1). 1 0 Z j Xj=1

The resolvent satisfies the equations of Section 2 except that (2.7) is replaced by

∞ ∗ Nλ(y, z)= pj(y)qj(z) /(λj − λ). (3.8) Xj=1

The Fredholm determinant is again given by (2.8). If only a finite number of eigenvalues are non- zero, the kernel N(y, z) is said to be degenerate. (For example this holds if µ puts weight only at n points.) If R = 1, that is,

|λ1| < |λj| for j > 1, (3.9) then as n →∞, n+1 n −1 n+1 n −1 N f(y)/N f(y) → λ1 , f(y)N /f(y)N → λ1 .

This is one way to obtain the first eigenvalue λ1 arbitrarily closely. Another is to use

−1 λ = sup{ gN hdµ : ghdµ = 1} if λ1 > 0, (3.10) 1 Z Z −1 λ = inf{ gN hdµ : ghdµ = 1} if λ1 < 0. (3.11) 1 Z Z

6 The maximising/minimising functions are the first eigenfunctions g = g1, h = h1. These are unique up to a constant multiplier if (3.9) holds. If λ1 is known, one can use

n ∗ ∗ n ∗ (λ1N ) f(y) → p1(y) q fdµ, f(y) (λ1N ) → q1(y) f p1dµ, Z 1 Z

s for any function f :Ω → C , to approximate p1(y),q1(y). One may now repeat the procedure on the operator N1 corresponding to

∗ N1(y, z)= N(y, z) − ν1p1(y)q1(z) to approximate λ2,p2(y),q2(z) assuming that the next eigenvalue in magnitude, λ2, has multiplicity 1. For further details see Withers (1974, 1975, 1978) and references.

4 Fredholm theory for non-diagonal Jordan form

Non-diagonal Jordan form for matrices For N 6= N ∗ a matrix in Cs×s, its general Jordan form is

r −1 N = P JP where J = diag(J1, · · · , Jr), Jj = Jmj (λj), mj = s, Xj=1 λ 1 0 · · · 0 0 λ 1 · · · 0 J (λ) = λI + U =   , (4.1) m m m · · ·   0 0 0 · · · λ   for some matrix P , and Um is the m × m matrix with 1s on the superdiagonal and 0s elsewhere:

(Um)jk = δj,k−1.

(See [1] for example. If N and its eigenvalues are real, then P can be taken as real.) So for n ≥ 1,

n n −1 n n n N = P J P where J = diag(J1 , · · · , Jr ).

By the Binomial Theorem,

n n n n−a a a J (λ) = λ U and (U ) = δ − . m a m m jk j,k a Xa=0 m So Um = 0. For example

n n−1 − λ nλ J (λ)n = λnI + nλn 1U = . 2 2 2  0 λn 

So N n can be expanded in block matrix form

r n n ck (N )jk = Pjc Jc P Xc=1

7 where we partition P and its inverse as

−1 jk P = (Pjk : j, k = 1, · · · , r), P = (P : j, k = 1, · · · , r)

jk mj ×mk with elements Pjk and P matrices in C . Alternatively setting

∗ −1 Q = P , (P1, · · · , Pr)= P, (Q1, · · · ,Qr)= Q,

s×mj with Pj,Qj ∈ C , we have

r n n ∗ n ∗ N = P J Q = PjJj Qj , (4.2) Xj=1 r ∗ ∗ ∗ ∗ PjQj = PQ = Is = Q P = (Qj Pk) Xj=1 so that ∗ ∗ mj ×mk Qj Pj = Imj , Qj Pk = 0 ∈ C if j 6= k.

P can be obtained as follows. Let pjk be the kth column of Pj for k = 1, · · · ,mj. Then

NP = P J ⇒ NPj = PjJj ⇒ Npjk = λjpjk + pj,k−1 where pj0 = 0. (4.3)

So one first obtains pj1, the right eigenvector of N, then pj2, · · · ,pjmj . This is called the Jordan chain. Q can either be obtained by inverting P or using ∗ ∗ ∗ ∗ ∗ ¯ N Q = QJ ⇒ N Qj = QjJj ⇒ N qjk = λjqjk + qj,k+1 where qj,mj +1 = 0. (4.4) ∗ So one first computes qj,mj , the right eigenvector of N then qj,mj −1, · · · ,qj1. For large n and λ 6= 0, n − − J (λ)n = λn m+1[U m 1 + O(1)] n m − 1 m m−1 and Um is a matrix of 0’s except for a 1 in its upper right corner. So if (3.2) holds and

R m1 = · · · = mM > max mj, j=M+1 then

n − − (N n) = rn M+1[D + O(n 1)] jk M − 1 1 n M M i(n−M+1)θc M−1 ck i(n−M+1)θc M−1 ck where Dn = Pjc e UM P = Pjc e UM P = O(1). Xc=1 Xc=1 See Withers and Nadarajah (2008) for more details. Non-diagonal Jordan form for functions Now consider N :Ω2 → Cs×s. Suppose that µ is a σ-finite measure on Ω and that N is not Hermitian, that is N(y, z)∗ 6= N(z,y). Its Jordan form is

−1 N(y, z)= P (y)JP (z) = where J = diag(J1, J2, · · · ), Jj = Jmj (λj) (4.5)

8 s×∞ for P (y):Ω → C and Jm(λ) of (4.1) above. So partitioning −1 jk P (y) = (Pjk(y) : j, k = 1, 2, · · · ), P (z) = (P (z) : j, k = 1, 2, · · · ),

jk mj ×mk with elements Pjk(y) and P (z) matrix functions in Ω → C , we can partition the nth n−1 iterated kernel, Nn(y, z)= N N(y, z) as ∞ n ck [Nn(y, z)]jk = Pjc(y) Jc P (z). Xc=1 Alternatively setting ∗ −1 Q(z) = P (z) , (P1(y), P2(y), · · · )= P (y), (Q1(z),Q2(z), · · · )= Q(z),

s×mj with Pj(y),Qj(z):Ω → C , we have ∞ n ∗ n ∗ Nn(y, z) = P (y)J Q(z) = Pj(y)Jj Qj(z) . (4.6) Xj=1

P (y) can be obtained as follows. Let pjk(y) be the kth column of Pj(y) for k = 1, · · · ,mj. Then

N P (y)= P (y)J ⇒N Pj(y)= Pj(y)Jj ⇒N pjk(y)= λjpjk(y)+ pj,k−1(y) (4.7) where pj0(y) = 0. So one first obtains pj1(y), the right eigenfunction of N, then pj2(y), · · · ,pjmj (y). Q(z) can either be obtained by inverting P (z) or using ∗ ∗ ∗ ∗ ∗ ¯ N Q(z)= Q(z)J ⇒N Qj(z)= Qj(z)Jj ⇒N qjk(z)= λjqjk(z)+ qj,k+1(z) (4.8) ∗ where qj,mj+1(z) = 0. So one first computes qj,mj (z), the right eigenfunction of N then qj,mj −1(z), · · · ,qj1(z). So if (3.2) holds and R m1 = · · · = mM > max mj, j=M+1 then

n − − (N (y, z)) = rn M+1(D (y, z)+ O(n 1)) n jk M − 1 1 jkn M i(n−M+1)θc M−1 ck where Djkn(y, z) = e Pjc(y)UM P (z)= O(1) Xc=1

M i(n−M+1)θc ck has (a, b) element c=1 e [Pjc(y)]a1[P (z)]Mb. P Example 4.1

5 The SVD for functions of two variables

The SVD for matrices s1×s2 Suppose that N ∈ C . That is, N is a s1 × s2 complex matrix. Denote its complex conjugate transpose by N ∗. Its SVD is r ∗ ∗ ∗ ∗ N = PDQ = θjpjqj where P P = I, QQ = I, r = min(s1,s2), (5.1) Xj=1 s1×s1 s2×s2 P = (p1, · · · ,ps1 ) ∈ C , Q = (q1, · · · ,qs2 ) ∈ C , θ1 ≥···≥ θr > 0

9 and for s1 = s2, s1 >s2, s1

So for s1 >s2, 2 ∗ Λ 0 ∗ DD = , D D = Λ2  0 0  and for s1

∗ Nqj = θjpj,N pj = θjqj,

∗ for r < j ≤ s1, Nqj = 0, and for r < j ≤ s2,N pj = 0. Also since ∗ ∗ ∗ ∗ NN P = P DD ,N NQ = QD D,

∗ 2 the pj is a right eigenvector of NN with eigenvalue θj (or 0 if r < j ≤ s1) and the qj is a right ∗ 2 eigenvector of N N with eigenvalue θj (or 0 if r < j ≤ s2). So (or by Section 2),

r r ∗ n 2n ∗ ∗ n 2n ∗ (NN ) = θj pjpj , (N N) = θj qjqj for n ≥ 1, Xj=1 Xj=1 r r ∗ n 2n+1 ∗ ∗ n ∗ 2n+1 ∗ (NN ) N = θj pjqj , (N N) N = θj qjpj for n ≥ 0. (5.2) Xj=1 Xj=1

These do not depend on the vectors {pj,qj, j ≥ r}.

So if θ1 = · · · = θM >θM+1, then we have the approximations for n ≥ 0,

M M ∗ n 2n ∗ 2n ∗ n 2n ∗ 2n (NN ) = θ1 pjpj + O(θM+1), (N N) = θ1 qjqj + O(θM+1) for n ≥ 1, Xj=1 Xj=1 r r ∗ n 2n+1 ∗ 2n+1 ∗ n ∗ 2n+1 ∗ 2n+1 (NN ) N = θ1 pjqj + O(θM+1 ), (N N) N = θ1 qjpj + O(θM+1 ). (5.3) Xj=1 Xj=1

But s1 s2 ∗ 0 ∗ ∗ 0 ∗ Is1 = (NN ) = pjpj , Is1 = (N N) = qjqj . Xj=1 Xj=1

If s1 = s2 and N is non-singular, its inverse is − − ∗ N 1 = QΛ 1P .

However unlike Jordan form, the SVD does not give a nice form for powers of N. Now suppose Ω ⊂ Rp and that µ is a σ-finite measure on Ω. Consider a function N(y, z):Ω2 → Cs1×s2 .

10 The equations ∗ N q(y)= θp(y), N p(z)= θq(z), have a countable number of solutions, say {θj,pj(y),qj(z), j ≥ 1} satisfying

∗ ∗ p pkdµ = q qkdµ = δjk. Z j Z j

The singular values {θj} may be taken as real, non-negative and non-increasing. (For convenience ∗ ∗ we have included θj = 0.) {pj(y)} and {qj(y)} are the right eigenfunctions of NN and N N 2 respectively, with eigenvalues {θj }. Also in L2(µ × µ) ∞ ∗ N(y, z)= θjpj(y)qj(z) . (5.4) Xj=1

If N is real, then so are {pj,qj}. By (5.4 ), for n ≥ 0, ∞ ∗ n 2n+1 ∗ (NN ) N(y, z) = θj pj(y)qj(z) , Xj=1 ∞ ∗ n ∗ 2n+1 ∗ (N N ) N(y, z) = θj qj(z)pj(y) , Xj=1 ∞ ∗ ∗ n 2n+2 ∗ N (NN ) N(y, z) = θj qj(y)qj(z) , Xj=1 ∞ ∗ n ∗ 2n+2 ∗ N (N N ) N(y, z) = θj pj(z)pj(y) , Xj=1

∞ ∗ n 2n ∗ (NN ) p(y) = θ pj(y) p pdµ, j Z j Xj=1 ∞ ∗ n 2n ∗ (N N ) q(z) = θ qj(z) q qdµ, j Z j Xj=1 ∞ ∗ n 2n+1 ∗ (NN ) N q(y) = θ pj(y) q qdµ, j Z j Xj=1 ∞ ∗ n ∗ 2n+1 ∗ (N N ) N p(y) = θ qj(y) p pdµ, j Z j Xj=1 ∞ ∗ ∗ n ∗ n 2n+2 trace N (NN ) N(y, z)|z=ydµ(y) = trace N (N N ) N(y, z)|z=ydµ(y)= θ . Z Z j Xj=1

So if θ1 = · · · = θM >θM+1, then we have approximations such as ∞ ∗ n 2n ∗ 2n (NN ) p(y)= θ pj(y) p pdµ + O(θ ) 1 Z j M+1 Xj=1 ∗ ∗ ∗ if pj(y) pj pdµ = O(1) for j > M. So for iterations of N N or NN the most important parameter is the largestR singular value.

11 References

[1] Kotz,S., Balakrishnan, N. and Johnson, N.L. (2000) Continuous multivariate distributions, 1, 2nd edition, Wiley, New York.

[2] Withers, C. S. (1974) Mercer’s Theorem and Fredholm resolvents. Bull. Austral. Math. Soc., 11, 373-380.

[3] Withers, C. S. (1975) Fredholm theory for arbitrary measure spaces. Bull. Austral. Math. Soc., 12, 283-292.

[4] Withers, C. S. (1978) Fredholm equations have uniformly convergent solutions. Jnl. of . Math. Anal. and Applic., 64, 602-609.

[5] Withers, C. S. (2000) A simple expression for the multivariate Hermite polynomials. Statistics and Probability Letters, 47, 165-169.

[6] Withers, C. S. and Nadarajah, S. (2008) The nth power of a matrix and approximations for large n. Preprint http://arxiv.org/abs/0802.0502

12