Application and Generalization of Eigenvalues Perturbation Bounds for Hermitian Block Tridiagonal Matrices

The paper 
contains two parts. First, by applying the results about the eigenvalue perturbation 
bounds for Hermitian block tridiagonal matrices in paper [1], we obtain a new 
efficient method to estimate the perturbation bounds for singular values of 
block tridiagonal matrix. Second, we consider the perturbation bounds for eigenvalues 
of Hermitian matrix with block tridiagonal structure when its two adjacent 
blocks are perturbed simultaneously. In this case, when the eigenvalues of the 
perturbed matrix are well-separated from the spectrum of the diagonal blocks, 
our eigenvalues perturbation bounds are very sharp. The numerical examples 
illustrate the efficiency of our methods.


Introduction
There are many known results about eigenvalue perturbation bounds of Hermitian matrices. See example [2][3][4][5]. Among them, one well-known theory is the following result.
Theorem 2.1 [2]. Let A and A E + be n-by-n Hermitian matrices. Let i λ and ˆi λ denote the ith smallest eigenvalues of A and A E + , respectively. Then for 1, 2, , i n =  , we have where all the eigenvalues of A and A E + are indexed in ascending order. This is the Weyl's theorem, which is one of the most classic eigenvalue perturbation theories. When the perturbation matrix E is an arbitrary Hermitian matrix, the bounds obtained by Weyl's theorem can be very small. However, for Hermitian matrices with special sparse structures such as block tridiagonal Hermitian matrix, the Weyl's theorem may not be the best choice. For this reason, [1] The natural questions are that whether the above results can be used to estimate the perturbation bounds for singular values of a block tridiagonal matrix, and how to get the eigenvalues perturbation bounds when two adjacent blocks of the matrix A in the formula (1.2) are perturbed simultaneously. If we apply the results above repeatedly, we can obtain a weaker upper bounds. Inspired by these questions, in this paper, we expect to obtain the perturbation bounds for singular value of a block tridiagonal matrix. Further, we give a new idea to obtain the eigenvalues perturbation bounds by directly using the bounds of eigenvector elements rather than applying the results in [1] repeatedly.
The structure of this paper is organized as follows. In Section 2, we provide preliminaries to outline our basic idea of deriving eigenvalue perturbation bounds via bounding eigenvector components [1]. In Section 3, we present a new approach to estimate the perturbation bounds for the singular values of the block tridiagonal matrix via applying the ideas in paper [1]. In Section 4, we consider the case which the sth block and ( ) 1 th s + block of the matrix A are perturbed simultaneously and present a new perturbation bound of the i smallest eigenvalue i λ . Further, we discuss the eigenvalue perturbation bounds when the first s blocks of A are perturbed simultaneously and provide an algorithm to estimate the bounds. In Section 5, we present a numerical example to show the efficiency of our approach.
Notations. Let 2 ⋅ denote the matrix spectrum norm.

Preliminaries
For simplicity, the eigenvalues that we mention in this paper are all simple eigenvalues. We need the following conclusion about the partial derivative of simple eigenvalue of Especially, the perturbation matrix E has the special structure. For example, the perturbation matrix E has the form as the matrix s E whose block elements are zero except for the sth block. Moreover if ( ) Yuji Nakatsukasa [1] has derived the eigenvalues perturbation bounds for the case (1.2) with this idea. In the following, we shall describe in detail how this idea can be exploited to derive perturbation bounds of singular values for block tridiagonal matrix, and how this idea is expanded to derive eigenvalue perturbation bounds for our cases. Note

Singular Value Perturbation Bounds
In this section, we use the results in paper [1]

2 × 2 Case
Firstly, for the 2 By Jordan-Wielandt theorem[2-Theorem I.4.2], we know that the eigenvalues of the matrix 4 4 A ×  are i σ ± , where 1 i m n k ≤ ≤ + + +  . The same statement holds for 4 4 E ×  . Permuting the rows and columns of the matrix 4 4 A ×  appropriately, we can get that the matrix 4 4 A ×  is similar to Obviously, the matrix 11 21 4 4 21 22 So it is natural that we can apply the result of [1-Theorem 3.2] to get the conclusion. 

3 × 3 Case
Secondly, we study the perturbation bounds for singular values of 3 3 × case. Let be the singular values of 3 3 A × and 3 3

3
A E × × + , respectively. Similar to the discussion above, by permuting the rows and columns of 3 3 appropriately, we can get that the matrix 3 3 Obviously, both 6 6 A × and 6

n × n Case
Further, we gradually consider the general n n × case and extend above statements to the n n × block tridiagonal matrices. Let In what follows, we give an example to illustrate the singular values perturbation bounds obtained by our results.
Example 3.1. Consider the 4 4 × matrices A and E represented by

Eigenvalue Perturbation Bounds
On the basis of conclusions of the paper [1], in this section we study eigenvalue perturbation bounds of block tridiagonal matrix for the cases where two adjacent blocks of A are perturbed and the first s blocks of A are perturbed by the perturbation matrix 1, ,s E  .

Two Adjacent Blocks of A Being Perturbed
In this subsection, we discuss eigenvalue perturbation bounds when two adjacent blocks of A are perturbed. In other words, we consider the matrices in the following form Similar to discussion of the paper [1], we need the following assumption. where the right inequality follows from Assumption 1. Continuously, the second block component of

The First s Blocks of A Being Perturbed
In this subsection, we gradually consider the bounds of eigenvalues of the matrix A , whose the first s blocks are perturbed simultaneously. In other words, we consider the perturbation matrix where s is a positive integer.Let . If i λ satisfies the Assumption 1, through the similar discussion as above, we can derive a similar conclusion for calculating the eigenvalue perturbation bounds. For simplicity, we don't repeat the proof here. The

Numerical Example
In this section, we use the following example to illustrate the validity of our method and to show the advantage of the our method over the method proposed in [1]. where all the elements of E are zero except for the 900th and 901th off diagonal, which are 1 ( ) . ., 900 i e s = . Note that none of the off-diagonals is negligibly small. We focus on i λ (the ith smallest eigenvalue of A ) for 1, ,10 i =  , which are smaller than 10. For such i λ we have 87 =  , and give bounds for 1, ,10 i =  with our method. The results are outlined in Table 1.
Meanwhile, we use the method in the paper [1] to give the perturbation bounds for 1, ,10 i =  . The results are outlined in Table 2.
Further, we partition the matrix A E + as in the (5.1) again so that the block size is one except for the 900th block, which is 2-by-2 matrix (i.e., s = 900). Using the method in the paper [1] we have the following perturbation bounds for 1, ,10 i =  , which are outlined in Table 3. Obviously, comparing the Table 1 with the Table 2, we can see that our method saves CPU times and improves the perturbation bounds. In addition, comparing the Table 1 with the Table 3, although our CPU time is close to the CPU time in Table 3, we see that the perturbation bounds are also improved . So we can say that our method is efficient and improved.

Conclusion
We have obtained a new efficient method to estimate the perturbation bounds for singular values of block tridiagonal matrix. Further, under the bases of the paper [1], we present a new conclusion for estimating the perturbation bound when the sth block and ( ) 1 th s + block of the matrix A are perturbed simultaneously and