Absolute Variation of Ritz Values, Principal Angles, and Spectral Spread

Abstract

Let A be a d × d complex self-adjoint matrix, X , Y ⊂ C d be k-dimensional subspaces and let X be a d × k complex matrix whose columns form an orthonormal basis of X ; that is, X is an isometry whose range is the subspace X . We construct a d × k complex matrix Y r whose columns form an orthonormal basis of Y and obtain sharp upper bounds for the singular values s(X * AX − Y * r A Y r ) in terms of submajorization relations involving the principal angles between X and Y and the spectral spread of A. We apply these results to obtain sharp upper bounds for the absolute variation of the Ritz values of A associated with the subspaces X and Y, that partially confirm conjectures by Knyazev and Argentati.

The spectral spread of Hermitian matrices

“…We expect that our results will have applications in matrix perturbation bounds for Hermitian matrices, obtained from a (differential) geometrical perspective. Indeed, in [18] we have already applied the results herein and obtained some inequalities related to the bounds in Eq. (3) (see [15]) using a geometrical approach.…”

“…( 5) together with some of its equivalent forms allow one to develop inequalities related to Eq. (3) (see [18]). In the last section of the paper we show the equivalence of the inequalities in Eqs.…”

Norm inequalities for the spectral spread of Hermitian operators

“…The next result is formally analogous to [24, Theorem 3.1.] (which played a central role in [25]). We point out that the spectral spread of a self-adjoint matrix A differs from the spectral spread of the self-adjoint compact operator à obtained from embedding A as a finite rank operator (see Section 5.1).…”

“…It turns out that all these singular value inequalities fail in this more general setting. Motivated by our previous work [24,25] (in the finite dimensional case) we first introduce a new notion that we call the spectral spread of a self-adjoint operator that lies in the algebra A = K(H) + C I formed by compact perturbations of multiples of the identity operator acting on H. Then, we obtain inequalities in terms of submajorization (which can be regarded as inequalities for unitarily invariant norms) with respect to the spectral spread, in the general context of compact self-adjoint operators. We regard these new inequalities as natural substitutes of the singular value inequalities mentioned above (that are stronger, but only valid for positive compact operators).…”

Norm inequalities for the spectral spread of Hermitian operators

“…Motivated by Knyazev-Argentati's work [25] and our previous works [26][27][28] we introduce the spectral spread of selfadjoint operators acting on a separable infinite-dimensional Hilbert space . In order to describe the spectral spread of a self-adjoint operator 𝐴 ∈ 𝐵(), we consider its spectral scale 𝜆(𝐴) = (𝜆 𝑖 (𝐴)) 𝑖∈ℤ 0 where ℤ 0 = ℤ ⧵ {0} (see Definition 2.1) defined by a "min-max" method.…”