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Linear and Multilinear Algebra
cauchy-lmatrevisedf
Linear and Multilinear Algebra Vol. 00, No. 00, Month 200x, 1–23
The Cauchy interlacing theorem in simple Euclidean Jordan algebras and some consequences M. Seetharama Gowdaa∗ and J. Tao a
b
Department of Mathematics and Statistics, University of Maryland, Baltimore County, Baltimore, MD 21250; b Department of Mathematical Sciences, Loyola College in Maryland, Baltimore, Maryland 21210 (Received 12 August 2008) In this article, based on the min-max theorem of Hirzebruch, we formulate and prove the Cauchy interlacing theorem in simple Euclidean Jordan algebras. As a consequence , we relate the inertias of an element and its principal components and extend some well known matrix theory theorems and inequalities to the setting of simple Euclidean Jordan algebras.
Keywords: Euclidean Jordan algebras, quadratic representations, min-max theorem of Hirzebruch, Cauchy interlacing theorem, Schur’s theorem, Hadamard’s inequality, Fan’s trace inequality AMS Subject Classification: 15A33; 17C20; 17C55
1.
Introduction
In matrix theory, the well known Cauchy’s interlacing theorem states that if A is a (complex) Hermitian matrix of size n × n and B is a leading principal submatrix of A of size k × k, then λ↓i (A) ≥ λ↓i (B) ≥ λ↓n−k+i (A)
(i = 1, 2, . . . , k),
where λ↓1 (A), λ↓2 (A), · · · , λ↓n (A) denote the eigenvalues of A written in the decreasing order (with a similar notation for B). This result has many interesting consequences. For example, given any two (complex) Hermitian matrices A and B, we have the following: (1) If B is a principal submatrix of A, then π(B) ≤ π(A) and ν(B) ≤ ν(A), where π(A) and ν(A) denote the number of positive and negative eigenvalues of A respectively (and likewise for B); (2) A is positive definite (semidefinite) if and only if all its leading principal minors are positive (respectively, nonnegative); (3) Schur’s theorem: diag(A) ≺ λ(A), where diag(A) and λ(A) refer to the diagonal of A and the vector of eigenvalues of A (written in the decreasing order), respectively; (4) Hadamard’s inequality: If A is positive semidefinite with diagonal elements a11 , a22 , . . . , ann , then det(A) ≤ a11 a22 · · · ann ; ∗ Email:
[email protected]
ISSN: 0308-1087 print/ISSN 1563-5139 online c 200x Taylor & Francis
DOI: 10.1080/03081080xxxxxxxxx http://www.informaworld.com
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Linear and Multilinear Algebra
cauchy-lmatrevisedf
M. Seetharama Gowda and J. Tao
(5) Fan’s inequality: trace(AB) ≤
Pn 1
λ↓i (A)λ↓i (B).
In this paper, we extend Cauchy’s interlacing theorem to simple Euclidean Jordan algebras and deduce analogs of the above statements in the setting of simple Euclidean Jordan algebras. Let (V, ◦, h·, ·i) be a simple Euclidean Jordan algebra of rank r (see Section 2 for definition). For any nonzero idempotent c in V , let Pc (x) := 2c ◦ (c ◦ x) − c2 ◦ x define the quadratic representation of c. Also, let V (c, 1) := {x ∈ V : c ◦ x = x}. Then V (c, 1) is a simple subalgebra of V and for any z ∈ V , z := Pc (z) ∈ V (c, 1). Denoting the eigenvalues of any z in V by the decreasing sequence λ↓1 (z), λ↓2 (z), · · · , λ↓r (z) and similarly for z in V (c, 1), we state the Cauchy interlacing inequalities in simple Euclidean Jordan algebras: λ↓i (z) ≥ λ↓i (z) ≥ λ↓r−k+i (z)
(i = 1, 2, . . . , k),
where k is the rank of V (c, 1). To recover the matrix theoretic result, one has to take V = Herm(C n×n ) (the space of all n × n complex Hermitian matrices with trace inner product and Jordan product defined by X ◦ Y := 21 (XY + Y X)) and the idempotent
Ik×k 0 C := , 0 0 where Ik×k is the identity matrix of size k × k. Our proof of the interlacing inequalities/theorem is based on the min-max theorem of Hirzebruch [13] which generalizes the well-known Courant-Fischer-Weyl min-max theorem (see [3]) to simple Euclidean Jordan algebras. As a consequence of the above inequalities, we state analogs of items/statements (1) − (5) above for Euclidean Jordan algebras. We also state a result that describes the inertia of an element in a Euclidean Jordan algebra when a specific Peirce decomposition of that element is known. In the case of simple algebras Herm(Rn×n ) (the space of all n × n real symmetric matrices) and Herm(C n×n ), our results reduce to the classical results. When specialized to Herm(Hn×n ) (the space of all n × n quaternionic Hermitian matrices) our interlacing inequalities reduce to a result of Tam [20] who proved the interlacing inequalities in the setting of Lie algebras. Our results are new in the (only other) simple algebras Ln (the so called Jordan spin algebra) for n > 2 and Herm(O3×3 ) (the space of all 3 × 3 octonionic Hermitian matrices). (We remark that in the setting of Herm(O3×3 ) our results are for ‘spectral’ eigenvalues, as there is a difference between such eigenvalues and real eigenvalues, see Sec. 2.2.)
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2.
3
Euclidean Jordan Algebras
We assume that the reader is familiar with the basic Euclidean Jordan algebra theory which can be found, for example, in Faraut and Kor´anyi [7]. For brief introductions, see Schmieta and Alizadeh [19], and Gowda, Sznajder, and Tao [10]. Let the triple (V, ◦, h·, ·i) denote a Euclidean Jordan algebra, where (V, h·, ·i) is a finite dimensional inner product space over R (the field of real numbers) and (x, y) 7→ x ◦ y : V × V → V is a bilinear mapping satisfying the following conditions for all x and y: x ◦ y = y ◦ x, x ◦ (x2 ◦ y) = x2 ◦ (x ◦ y), and hx ◦ y, zi = hy, x ◦ zi. We denote the unit element of V by e. In V , the cone of squares {x2 : x ∈ V } (which is a symmetric cone) is denoted by K. For an element z ∈ V , we write z ≥ 0 (z > 0)
if and only if z ∈ K (z ∈ K o ),
where K o denotes the interior of K. A Euclidean Jordan algebra is said to be simple if it is not the direct product of two (non-trivial) Euclidean Jordan algebras. The classification theorem (Chapter V, Faraut and Kor´anyi [7]) says that every simple Euclidean Jordan algebra is isomorphic to the (Jordan spin) algebra Ln or to the algebra of all n × n real/complex/quaternion Hermitian matrices with (real) trace inner product and X ◦ Y = 21 (XY + Y X) or the algebra of all 3 × 3 octonion Hermitian matrices with (real) trace inner product and X ◦ Y = 12 (XY + Y X). Furthermore, the structure theorem, see (Chapters III and V, Faraut and Kor´anyi [7]) says that any Euclidean Jordan algebra is a (Cartesian) product of simple Euclidean Jordan algebras. Let V be a Euclidean Jordan algebra of rank r. An element c ∈ V is called an idempotent if c2 = c; it is a primitive idempotent if it is nonzero and cannot be written as the sum of two nonzero idempotents. A finite set {e1 , . . . , er } is said to be a Jordan frame in V if each ei is a primitive idempotent in V , ei ◦ej = δij ei for all i, j = 1, 2, . . . , r and e1 + e2 + · · · + er = e. (These conditions imply hei , ej i = 0 for all i 6= j = 1, 2, . . . , r.) The spectral theorem says that for each element x in V , there exists a Jordan frame {e1 , . . . , er } and real numbers λi (i = 1, 2, . . . , r) such that x = λ1 e1 + · · · + λr er .
(1)
The above expression is the spectral decomposition (or the spectral expansion) of x. The real numbers λi (also written as λi (x)) are the (spectral) eigenvalues of x; these are uniquely defined, even though the Jordan frame corresponding to x need not be unique. For any x ∈ V given by (1), we define the trace and determinant of x by trace(x) := λ1 + λ2 + · · · + λr and det(x) := λ1 λ2 · · · λr .
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We also define the inertia of x by In(x) = (π(x), ν(x), δ(x)), where π(x), ν(x), and δ(x) are, respectively, the number of eigenvalues of x which are positive, negative, and zero, counting multiplicities. Clearly, π(x) + ν(x) + δ(x) = r for all x. We note that hu, vit := trace(u ◦ v) defines another inner product on V so that (V, ◦, h·, ·it ) is also an Euclidean Jordan algebra. Fix a Jordan frame {e1 , e2 , . . . , er } in a Euclidean Jordan algebra V . For i, j ∈ {1, 2, . . . , r}, define the Peirce spaces Vii := {x ∈ V : x ◦ ei = x} = Rei and when i 6= j, 1 Vij := {x ∈ V : x ◦ ei = x = x ◦ ej }. 2 Then we have the following. Theorem 2.1 : (Theorem IV.2.1, Faraut and Kor´ anyi [7]) The space V is the orthogonal direct sum of spaces Vij (i ≤ j). Furthermore, Vij ◦ Vij ⊂ Vii + Vjj , Vij ◦ Vjk ⊂ Vik if i 6= k, and Vij ◦ Vkl = {0} if {i, j} ∩ {k, l} = ∅. Thus, given a Jordan frame {e1 , e2 , . . . , er }, we can write any element x ∈ V as x=
r X i=1
xi ei +
X
xij
(2)
i 0. (Note that the eigenvalues and determinants vary continuously on the element, see e.g., [11].) For the converse result(s), we induct on the rank of V . The result is obviously true for r = 1. So, assume the result for any simple algebra of rank r−1. For the given Jordan frame {e1 , e2 , . . . , er }, we consider the subalgebra V (r−1) which is simple (according to Item (vi) in Prop. 4.1) and of rank r − 1. Let z = Pe1 +e2 +···+er−1 (z). For ease of notation, let αi (βi ) denote the eigenvalues of z (respectively, z) given in the decreasing order. Then by the Cauchy interlacing theorem given above, we have α1 ≥ β1 ≥ α2 ≥ β2 · · · ≥ αr−1 ≥ βr−1 ≥ αr . Now suppose that ∆k (z) > 0 for all k = 1, 2, . . . , r. As e1 + e2 + · · · + el and e1 + e2 + · · · + er−1 operator commute, Pe1 +e2 +···+el Pe1 +e2 +···+er−1 = P(e1 +e2 +···+el )◦(e1 +e2 +···+er−1 ) = Pe1 +e2 +···+el for all l = 1, 2, . . . , r − 1, and hence we have ∆l (z) = ∆l (z) > 0 for all l = 1, 2, . . . , r − 1. By induction, z > 0 in V (r−1) which means that βi > 0 for all i = 1, 2, . . . , r − 1. It follows that αi > 0 for all i = 1, 2, . . . , r − 1. Since 0 < ∆r (z) = det(z) = α1 α2 · · · αr
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it follows that αr > 0. Hence all the eigenvalues of z are positive. This proves that z > 0. The case z ≥ 0 follows by considering z + εe for small positive ε and using the continuity of eigenvalues [11]. Given a vector x = (x1 , x2 , . . . , xr ) in Rr , we write x↓ := (x↓1 , x↓2 , . . . , x↓r ) for the vector obtained by rearranging the components of x in the decreasing order. For two vectors x = (x1 , x2 , . . . , xr ) and y = (y1 , y2 , . . . , yr ) in Rr , we say that x is majorized by y and write x ≺ y if k X 1
x↓i ≤
k X
yi↓
(k = 1, 2, . . . , r − 1)
1
and r X
x↓i
1
=
r X
yi↓ .
1
In matrix theory, the well known majorization theorem of Schur says that for a complex Hermitian matrix A, diag(A) ≺ λ(A), where diag(A) and λ(A) refer to the diagonal of A and the vector of eigenvalues of A (written in the decreasing order) respectively. We have a similar result in any simple Jordan algebra. Corollary 4.6: Let V be simple and {e1 , e2 , . . . , er } be any Jordan frame in V . Let a ∈ V with the Peirce decomposition given by a=
r X 1
ai ei +
X
aij .
i
