Location for the Left Eigenvalues of Quaternionic Matrix

The purpose of this paper is to locate and estimate the left eigenvalues of quaternionic matrices. We present some distribution theorems for the left eigenvalues of square quaternionic matrices based on the generalized Gerschgorin theorem and generalized Brauer theorem.


Introduction and symbols
Linear algebra over quaternion division algebra has drawn the attention of the researchers of mathematics, physics, and computer science widely.The research of matrices is continuously an important aspect of algebra problem over quaternion division algebra.Quaternion is mostly used in computer vision because of their ability to represent rotations in 4D spaces [Ran Ruisheng, 2006].It is also playing an important role in quantum physics and geostatics and so forth [Wang Qinggui, 1983; Zhang Gangshu, 1982].
In this paper we shall mainly adopt the notation and terminology in [Fuzhen Zhang, 2007], for convenience, recall it.As usual, R and C are the sets of the real and complex numbers respectively.We denote by H the set of real quaternions: . Let H n×n and H n×1 be the collections of all n × n matrices with entries in H and n-column vectors respectively.Let I n be the collections of all n × n unit matrices with entries in H.For X ∈ H n×1 , X T is the transpose of

Some definitions and lemmas
Before starting, we quickly review some basic definitions and lemmas.
Definition 2.1 (Junliang Wu, 2008) Suppose that a is a given real quaternion and is a positive real number.The set G = {z ∈ H : |z − a| ≤ r} is said to be a generalized spherical neighborhood with the center a and the radius r.Definition 2.2 (Fuzhen Zhang, 1997) Let A ∈ H n×n , a quaternion λ is said to be left eigenvalue of A if there exists nonzero quaternion column vector Zhang, 1997) Let A ∈ H n×n , a quaternion λ is said to be right eigenvalue of A if there exists nonzero quaternion column vector Lemma 2.1 (Fuzhen Zhang, 2007) (The generalized Gerschgorin Theorem) Let A = (a i j ) ∈ H n×n .Then all the left eigenvalues of A are located in the union of n generalized spherical neighborhoods Wu, 2008) (The generalized Gerschgorin Theorem) Let A = (a i j ) ∈ H n×n .Then all the eigenvalues of A are located in the union of n generalized spherical neighborhoods Lemma 2.3 (Junliang Wu, 2008).(The generalized Brauer Theorem) Let A = (a i j ) ∈ H n×n .Then all the left eigenvalues of A are located in the union of n(n−1) 2 generalized oval neighborhoods where

The location for the left eigenvalues of quaternionic matrix
Recently, the forms of the famous Gerschgorin Theorem and Brauer Theorem were studied over real quaternion division algebra (Fuzhen Zhang, 2007;Junliang Wu, 2008;Junliang Wu, 2008).They solved some distribution and estimation problems of left eigenvalues of quaternionic matrices.In this section, we will present some distribution theorems for the left eigenvalues of square quaternionic matrices based on the generalized Gerschgorin Theorem and Brauer Theorem.
Theorem 1.Let A = (a i j ) ∈ H n×n and let α ∈ [0, 1] be given.Then all the left eigenvalues of A are located in the union of n generalized spherical neighborhoods where α(A) is the set of the left eigenvalues of A, Proof.As the cases α = 0 and α = 1, by Lemma 2.1 and lemma 2.2, the result holds.Furthermore, we may assume that |a i j | > 0, because we may perturb A by inserting a small nonzero entry into any row in which the resulting matrix has an inclusion generalized spherical neighborhood that is larger than the generalized spherical neighborhood for A. And the result follows in the limit as the perturbation goes to zero.Now we suppose that AX = λX with 0 Let p = 1 α and q = 1 1−α , then 1 p + 1 q = 1.By Holder's inequality (page 147), it is not hard to see the following Using Cauchy-Schwarz inequality (Junliang Wu, 2008), we have . Since we do not know which i is appropriate to each left eigenvalue λ (unless we know the associated eigenvector, in which case we would know λ exactly and would not be interested in locating it), we can only conclude that λ lies in the union of n generalized spherical neighborhoods.Then the theorem is proved.2Theorem 2. Let A = (a i j ) ∈ H n×n .Then all the left eigenvalues of A are located in the union of n(n−1) 2 generalized oval neighborhoods where λ(A) is the set of the left eigenvalues of A, R i = Proof.Let λ be a left eigenvalue of A, then AX = λX, where 0 Obviously, if the other entries of quaternion vector X are zero, then the result holds naturally.If there are at least two entries of vector X are nonzero, then we let Using Cauchy-Schwarz inequality (Junliang Wu, 2008), we have Meanwhile, we also have Taking the produce of (3.1) and (3.2) permits us to eliminate the unknown ratios of component of x and obtain Thus, all the left eigenvalues of A are located in the union of n(n−1) 2 generalized oval neighborhoods.Then the proof is completed.2Theorem 3. Let A = (a i j ) ∈ H n×n .Then all the left eigenvalues of A are located in the union of n(n−1) where λ(A) is the set of the left eigenvalues of A, R i = Proof.Let λ be a left eigenvalue of A, then AX = λX, where 0 The proof is completed.2 Example.Let By generalized Brauer Theorem (Junliang Wu, 2008), we can obtain that all the left eigenvalues of A are located in the union of following generalized oval neighborhoods By Theorem 3, we can obtain that all the left eigenvalues of are located in the union of following generalized oval neighborhoods It is clear that the result of Theorem 3 is sharper than the generalized Brauer Theorem's.Therefore, Theorem 3 can obtain a good estimation for some quaternion matrices.Proof.According to the proof method of Theorem 3, we can prove Theorem 4, so we leave it to readers.

Theorem 4 .
Let A = (a i j ) ∈ H n×n .Then all the left eigenvalues of A are located in the union of n(n−1) 2 generalized oval neighborhoods λ(A) ⊂ G(A) = ∈ H : |z − a ii ||z − a j j | ≤ √ n − 1R i ( √ n − 1R i + |a ii − a j j |)},where λ(A) is the set of the left eigenvalues ofA, R i = n j=1 j i |a i j | 2 , i = 1, 2, • • • , n.