Vol. 34, No. 2, 1970

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An analysis of equality in certain matrix inequalities. I

William R. Gordon and Marvin David Marcus

Vol. 34 (1970), No. 2, 407–413
Abstract

In this paper we are concerned with analyzing the cases of equality in certain inequalities that relate the eigenvalues and main diagonal elements of hermitian matrices.

Let Er denote the r-th elementary symmetric function of k variables (E0 = 1). If H = (hτj) is an n-square positive semidefinite hermitian matrix with eigenvalues γ1 γn and if 1 r k n, then it is known that

Er(h11,⋅⋅⋅ ,hkk) ≧ Er(γ1,⋅⋅⋅ ,γk).
(1.1)

If r > 1 and at least r of h11,,hkk are positive then (1.1) can be equality if and only if there exists a permutation φ lSk such that

H = diag(γφ(1),⋅⋅⋅ ,γφ(k)) +Hn −k
(1.2)

where Hnk is (nk)-square and + denotes direct sum. Of course, if r = k = n then (1.1) is the Hadamard determinant theorem:

n∏
hii ≧ det(H).
i=1
(1.3)

If some hii = 0, then H is singular and (1.3) is equality. If hii > 0,i = 1,,n, then the condition (1.2) yields the well-known criterion for equality in (1.3), namely H = diag(h11,,hnn).

Mathematical Subject Classification
Primary: 15.55
Milestones
Received: 8 May 1969
Revised: 15 December 1969
Published: 1 August 1970
Authors
William R. Gordon
Marvin David Marcus