Probabilistic Perturbation Bounds for Invariant, Deflating and Singular Subspaces

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

In this paper, the authors consider the sensitivity of invariant subspaces, deflation subspaces and singular subspaces of matrices, and present some perturbation bounds. However, there are few new results and innovations in this paper. My recommendation is major revision. Some comments for your further improvement of the manuscript are given below.

1 What should you give for your contribution in the section 1

2 p.2 line 58 $\Delta a_{ij}=\|\delta A\|=$ means $\Deta A =\|\delta A\| J$, where $J$ is m-by-n matrix with each entry being 1. In this cane,  $...\|\delta A\|_F$  should  be $...\|\delta A\|$;

3 The application of probabilistic perturbation bound is not clear.

Author Response

Please see the attached file.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

This paper is a comprehensive review of the results on the perturbation bounds for invariant, deflating and singular subspaces for basic eigenvalue and singular value problems.

Probabilistic asymptotic perturbation bounds on the angles between perturbed and unperturbed subspaces are derived in a unified manner, based on the Markoff inequality. Numerical results illustrate that these probabilistic asymptotic bounds are less conservative than their deterministic counterparts.

The paper is well structured and clearly written. An extended list of relevant references is included.

I just can make few suggestions.

l. (line(s)) 177 states that M is nonsingular, but K := kron( I_n, T ) - kron( T^T, I_n ) is always singular, hence M is so. For a very simple examples, let T = diag( [ 1 2 ] ), in which case K = diag( [ 0 1 -1 0 ] ). But the matrix kron( I_m, T₁ ) - kron( T₂^T, I_p ), with m+p = n and T = blkdiag(T₁, T₂), is nonsingular if T₁ and T₂ have distinct spectra.  On the other hand, T and T^T have the same spectra.

l.262:  \cal{X} -> \tilde{\cal{X}}

l.297:  contains -> contain

l.333:  decreases -> decrease

l.332-333:  I cannot see on Fig.4 that the mean values decrease 8 times at P^ref = 90% and 16 times at P^ref = 90%. Maybe, this follows form the numerical results obtained. Some further explanation would be useful.

(77):  Who is \Sigma_n?  Would it be \sigma_n or is it diag( \sigma₁, \sigma₂, ..., \sigma_n )?

Who is \delta \Sigma_n in the line above l.439? 

l.530-531 should be united

References [3], [8], [13], [16], [31], [32], [37], [38] have authors' names written differently from all other references (which have all capital letters). Also, these references have journal name in Italics, but not authors' names (again, different from all other cases).

References [14], [15], [38] are not cited in the text.

Comments on the Quality of English Language

The English language is very well used, but I discovered few misuse, shown in my comments above.

Author Response

Please see the attached file.

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

Please see the attachment.

Comments for author File: Comments.pdf

Comments on the Quality of English Language

Please see the attachment.

Author Response

Please see the attached file.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

The author has addressed all comments in the new version, so I recommend this manuscript to publish in Axioms.

Reviewer 3 Report

Comments and Suggestions for Authors

The revised version is fine and I recommend this manuscript to be published.