A Simple Proof and Refinement of Wielandt’s Eigenvalue Inequality

A Simple Proof and Refinement of Wielandt’s Eigenvalue Inequality

Lutz D¨umbgen

published 1995 inStatist. Prob. Letters 25, 113-115

Abstract: Wielandt (1967) proved an eigenvalue inequality for partitioned sym- metric matrices, which turned out to be very useful in statistical applications. A simple proof yielding sharp bounds is given.

Keywords and phrases: eigenvalue inequality, partitioned matrix

Correspondence to: Lutz Duembgen, Institute of Mathematical Statistics and Actuarial Science, University of Bern, Sidlerstrasse 5, CH-3012 Bern, Switzerland;

e-mail: duembgen@stat.unibe.ch

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Let A∈R^p×p be a symmetric matrix of the form

A = B C

C⁰ D

!

withB ∈R^r×r,C ∈R^r×s and D∈R^s×s such that λr(B) > λ1(D);

generally λ1(E) ≥ λ2(E) ≥ · · · ≥ λq(E) denote the ordered eigenvalues of a sym- metric matrixE∈R^q×q. Wielandt (1967) showed that the eigenvalues ofAcan be approximated by the eigenvalues of B and D in the following sense:

0 ≤ λ_i(A)−λ_i(B) ≤ λ₁(CC⁰)

λi(B)−λ1(D) for 1≤i≤r and (1) 0 ≤ λ_j(D)−λ_r+j(A) ≤ λ₁(CC⁰)

λr(B)−λj(D) for 1≤j≤s.

These inequalities can be used to compute derivatives and pseudo-derivatives of eigenvalues. They are also very useful in statistical problems involving eigenvalues of random symmetric matrices; see Eaton and Tyler (1991, 1994). In my opinion the original proof, described in Eaton and Tyler (1991), is somewhat complicated.

The main ingredient seems to be the Courant-Fischer minimax representation λ_k(E) = max

V:dim(V)=k min

v∈V:v⁰v=1 v⁰Ev for 1≤k≤q, (2)

where V stands for a linear subspace of R^q; see section 1f.2 of Rao (1973). In this note (2) is used directly to derive the following refinement of (1):

Theorem. For1≤i≤r, 0 ≤ λi(A)−λi(B) ≤

s

(λi(B)−λ1(D))²

4 +λ1(CC⁰) − λi(B)−λ1(D)

2 ,

and for 1≤j≤s,

0 ≤ λ_j(D)−λ_r+j(A) ≤ s

(λ_r(B)−λ_j(D))²

4 +λ₁(CC⁰) − λ_r(B)−λ_j(D)

2 .

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Remark 1: Since q

α²/4 +β² − α/2 ≤ minⁿβ, β²/α^o ∀α, β >0, this result implies Wielandt’s bounds (1).

Remark 2: The upper bounds are sharp. For if p = 2 one can compute the eigenvalues ofA explicitly and obtains

λ₁(A)−λ₁(B) = λ₁(D)−λ₂(A) = s

(B−D)²

4 +C² − B−D 2 .

For general p one has to consider diagonal matrices B, D and suitable matrices C with only one nonzero coefficient.

Proof of the Theorem: One easily verifies that the asserted inequalities are invariant under the transformationA7→A−λ1(D)I, whereI is the identity matrix inR^p×p. Therefore one may assume without loss of generality thatλ₁(D) = 0.

For 1≤i≤r it follows from (2) that λ_i(A) ≥ max

V⊂R^r×{0}:dim(V)=i min

v∈V:v⁰v=1 v⁰Av = λ_i(B). (3) On the other hand, letW be ani-dimensional subspace of R^p such that

λi(A) = min

v∈W:v⁰v=1 v⁰Av.

Ifv∈R^p is written as v= (v₍₁₎⁰ , v₍₂₎⁰ )⁰ withv₍₁₎ ∈R^r and v₍₂₎ ∈R^s, then W₍₁₎ = {v₍₁₎:v∈W}

is an i-dimensional subspace of R^r. For if dim(W₍₁₎) < i, then w₍₁₎ = 0 for some unit vector w∈W, and

λ_i(A) ≤ w⁰Aw = w⁰₍₂₎Dw₍₂₎ ≤ 0,

which would contradict (3). Any unit vector v∈W can be written as v =

q

(1 +ρ)/2u₍₁₎+ q

(1−ρ)/2u₍₂₎ 3

for unit vectorsu₍₁₎ ∈W₍₁₎, u₍₂₎ ∈R^s and someρ∈[−1,1]. Then v⁰Av = (1 +ρ)u⁰₍₁₎Bu₍₁₎/2 +

q

1−ρ²u⁰₍₁₎Cu₍₂₎+ (1−ρ)u⁰₍₂₎Du₍₂₎/2

≤ (1 +ρ)u⁰₍₁₎Bu₍₁₎/2 + q

1−ρ^2qλ₁(CC⁰)

= u⁰₍₁₎Bu₍₁₎/2 +ρ, q

1−ρ² u⁰₍₁₎Bu₍₁₎/2 pλ1(CC⁰)

!

≤ u⁰₍₁₎Bu₍₁₎/2 +^q(u⁰₍₁₎Bu₍₁₎)²/4 +λ₁(CC⁰).

Consequently, since H(x) :=x/2 +^px²/4 +λ₁(CC⁰) is nondecreasing inx≥0, λi(A) ≤ min

u(1)∈W₍₁₎:u⁰₍₁₎u(1)=1 H(u⁰₍₁₎Bu₍₁₎)

= H min

u₍₁₎∈W₍₁₎:u⁰₍₁₎u₍₁₎=1u⁰₍₁₎Bu₍₁₎

!

≤ H(λ_i(B))

= λ_i(B) +^q(λ_i(B)−λ₁(D))²/4 +λ₁(CC⁰)−(λ_i(B)−λ₁(D))/2.

Thus the first part of the theorem is true, and the second half follows by replacing A with−A 2

References

Eaton, M.L. and D.E. Tyler (1991): On Wielandt’s inequality and its application to the asymptotic distribution of the eigenvalues of a random symmetric matrix.

Ann. Statist.19, 260-271

Eaton, M.L. and D.E. Tyler (1994): The asymptotic distribution of singular values with applications to canonical correlations and correspondence analysis. J.

Multivariate Anal. 50, 238-264

Rao, C.R. (1973): Linear Statistical Inference and Its Applications (2nd edition).

Wiley, New York

Wielandt, H. (1967): Topics in the Analytic Theory of Matrices. (Lecture notes prepared by R. R. Meyer) Univ. Wisconsin Press, Madison

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