1 Quasi-definite matrix

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1 1 Quasi-definite matrix The matrix H is a quasi-definite matrix, if there exists a permutation matrix P such that H qd P T H11 H HP = 1 H1, 1) H where H 11 and H + H1H 11 H 1 are positive definite. This allow us the following decomposition: I H11 H qd = H1H 11 I We are going to use the following notations H 1/ Γ = H1H 11, G = H + H 1H 11 H 1 ) H + H 1H 11 H 1 ) 1/ 11 I H 11 H 1 I, J =. ) I. 3) I The main result of this section will be the upper bound for the norm any unitary invariant norm) of J-unitary matrix F = F + F which simultaneously diagonalize the pair G I + Γ) I + Γ)G, J), that is F + F G I + Γ) Λ+ F I + Γ)G F+ F =, + Λ F J I F + F = I 4) Theorem 1.1. Let H qd be a quasi-definite matrix defined by 1)and let F be J-unitary matrix which simultaneously diagonalize the pair G I +Γ) I +Γ)G, J) as in 4). Then the following upper bound for F holds: F H 1H11 H H ) 11, 5) η η where λ i Λ, λ j Λ + λ i λ j λ i η = min j, 6) and where stands for any unitary invariant norm. 1

2 Rotation of eigensubspace for indefinite Hermitian pairs Finally in this section we will present perturbation bound for the Hermitian pair H, M), where H is quasi-definite matrix defined by 1) and M is positive definite, perturbed such that H, M) is again Hermitian indefinite pair, with H quasi-definite matrix and M positive definite. Let the quasi-definite matrix H and H be decomposed as H = GJG := I + Γ)G JG I + Γ), H = GJ G := I + Γ) G J G I + Γ), where Γ = H1H 11, G = and Γ = H 1 H 1/, G = H 11 Then we have the following theorem: H + H 1H 11 H 1 ) 1/ 11 1/ H 11 H + H 1, J = H 11 H 1 ) 1/ I. I Theorem.1. Let H, M) be a Hermitian pair and let H, M) be the perturbed pair. Let X = X 1 X and X = X1 X, be non-singular matrices which simultaneously diagonalize the pairs H, M) and H, M). Let H F = min 1H11 H H ) 11, F = min H 1 H 11 + H ) H 11, If then. H 1 H H1 H ) η η H H 1 + H ) H η G := G δg < 1, η M := M / δmm / < 1, sin Θ M RanX 1 ), Ran X 1 )) F F F RelGap Φ H η G RGap Φ M 1 ηm, 7)

3 where sin Θ M RanX 1 ), Ran X 1 )) is diagonal matrix with sines of canonical angles between RanX 1 ) and Ran X 1 ) on its diagonal given in the weighted M-inner product space, defined by??). Further, in 7) Φ M = M / δmm / F and Φ H = G δhg F and RelGap and RGap are defined by??) and??), respectively. Proof. The proof is similar to the proof of the Theorem??. 3 Numerical examples 3.1 Example Let H qd be quasi-definite matrix of the form H11 b H qd = b H 11, 8) where H 11 = diag1 : n), n = 1 is real symmetric positive definite matrix and b = b onesn), b R. We will considered the eigenvalue problem H qd x = λx. 9) Similarly as in previous example, problem 9) is equivalent to the problem where H11 bh 11 b + H 11 ) y = λ y = I H 11 b I I bh11 bh11 x. H11 b bh 11 Note that now we have eigenvector problem of the definite pair H, M) where H11 H = ) I H bh11 b 11 b and M = + H 11 bh11 bh11 ) + I y, 1) bh 11 ) + I. In next two experiments we will estimate the bound 7) for the eigenvector problems 9) and 1). As in previous example, we estimate the perturbation of an invariant subspace which corresponds to the first four smallest eigenvalues of the matrix H qd,and also for the matrix pair H, M). Experiment 3.1. We consider random perturbations δh qd, which satisfy δh) ij η H ij, 3

4 where η = 1 9. The exact perturbations gives sin ΘRanX 1 ), Ran X 1 ) F The quantities η G G δg and Φ H G δhg F are bounded by η G , Φ H Also,??) gives RelGap.4. The bound 5) for J-unitary matrices F and F gives F 1.974, F Now using the above, bound 7) yields sin ΘRanX 1 ), Ran X 1 ) F Experiment 3.. We consider random perturbations δh and δm, which satisfy δh) ij η H ij, δm) ij η M ij, where η = 1 9. The exact perturbations gives sin Θ M RanX 1 ), Ran X 1 ) F The quantities η G G δg and η M M / δmm / are bounded by η G , η M , and Φ H G δhg F, Φ M M / δmm / F are bounded by Φ H , Φ M Also,??) and??) gives RelGap.4, RGap.4. For J-unitary matrices F and F holds F = 1, F = 1. Now using the above, bound 7) yields sin Θ M RanY 1 ), RanỸ1) F

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