View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Linear Algebra and its Applications 323 (2001) 105–116 www.elsevier.com/locate/laa On left eigenvalues of a quaternionic matrix Liping Huang a ,∗,1,WasinSob,2 aInstitute of Mathematics and Software, Xiangtan Polytechnic University, Xiangtan, Hunan 411201, People’s Republic of China bDepartment of Mathematics and Computer Science, San Jose State University, San Jose, CA 95192-0103, USA Received 18 April 2000; accepted 15 August 2000 Submitted by C.-K. Li Abstract In this paper, we introduce the concept of left and right eigenvalues for a quaternionic matrix, and investigate their properties, quantities and relationship. © 2001 Elsevier Science Inc. All rights reserved. 1. Introduction Let R be the set of real numbers and H be the set of quaternions of the form a1 + a2i + a3j + a4k,whereai ∈ R and i2 = j2 = k2 = ijk =−1. For a = a1 + a2i + a3j + a4k ∈ H,leta = a1 − a2i − a3j − a4k be the conjugate of a,and √ q |a|= aa = a2 + a2 + a2 + a2 1 2 3 4 be the modulus of a.LetRe(a) = (a + a)/2bethereal part of a,andIm(a) = (a − a)/2betheimaginary part of a. Two quaternions a and b are similar if there exists a nonzero q ∈ H such that a = q−1bq. The similarity class of a is the set ∗ Corresponding author. E-mail addresses: [email protected] (L. Huang), [email protected] (W. So). 1 This research is partially supported by the National Natural Science Foundation of China and the NSF of Hunan Province. 2 This research was initiated during a visit to the Institute of Mathematics and Software in Xiangtan Polytechnic University. 0024-3795/01/$ - see front matter ( 2001 Elsevier Science Inc. All rights reserved. PII:S0024-3795(00)00246-9 106 L. Huang, W. So / Linear Algebra and its Applications 323 (2001) 105–116 {q−1aq: a/= 0}. Note that a is similar to b if and only if Re a = Re b and |a|=|b|. m×n Denote C the set of complex numbers of the form a1 + a2i,whereai ∈ R.LetX n be the set of all m × n matrices over a division ring X,andX be the set of vectors of n components over a division ring X,whereX = R, C or H. In this paper we are interested in the eigenvalues of a quaternionic matrix. Due to the noncommutativity of quaternions, there are two types of eigenvalues. Definition 1.1. Given A ∈ Hn×n, λ ∈ H is called a right eigenvalue of A if AX = Xλ for some nonzero X ∈ Hn. The set of distinct right eigenvalues is called the right spectrum of A, denoted σr (A). Definition 1.2. Given A ∈ Hn×n, λ ∈ H is called a left eigenvalue of A if AX = λX for some nonzero X ∈ Hn. The set of distinct left eigenvalues is called the left spectrum of A, denoted σl(A). It is known that right spectrum is always nonempty, and indeed right eigenvalues are well studied in literature [1,5]. On the contrary, left eigenvalues are less known. Zhang [9] commented that left spectrum is not easy to handle, and few results have been obtained. In his thesis, Siu [6] spent a whole chapter on discussing left and right eigenvalues of quaternionic matrices. This paper is a follow up of Siu’s effort and another response to Zhang’s call on the investigation of left eigenvalues of a quatern- ionic matrix. Note that A − λI is a singular matrix if and only if AX = λX for some nonzero X ∈ Hn. Hence left eigenvalues are also known as singular eigenvalues in literature. Cohn [2] raised the question whether left eigenvalue always exists. In a footnote of his paper Lee [5] doubted the existence of left eigenvalues in general, but he provided no example. Wood [8] used a topological method to confirm that the left eigenvalue always exists and at once gave a negative answer to Lee’s speculation. However Wood’s proof is existential and provided no algorithm for computing left eigenvalues. At the end of his paper, Wood suggested an algebraic approach to find left eigenvalues. He demonstrated that the left eigenvalues of a 2 × 2 matrix can be found by solving a quaternionic quadratic equation, but he did not actually solve the quadratic equation. Later So [7] reduced the existence of a left eigenvalue of a 3 × 3 quaternionic matrix to the existence of a solution of a generalized quaternionic poly- nomial of degree 3, which is guaranteed by the Fundamental Theorem of Algebra for quaternions [3]. It is still an open problem whether this algebraic approach works for general n × n matrices for n > 4. The rest of the paper is organized as follows. Section 2 concerns the computation of left eigenvalues. Using results from Huang and So [4] on quaternionic quadrat- ic formulas, we explain how to compute all left eigenvalues of a 2 × 2 matrix. In Section 3, we study the possible number of distinct left eigenvalues of a quaternion matrix. Section 4 is devoted to the conjecture that finiteness of both left and right spectra implies their equality. We also study those matrices with identical left and right spectra. L. Huang, W. So / Linear Algebra and its Applications 323 (2001) 105–116 107 2. Computation of left spectrum We begin this section with some useful observation about a general left spectrum. Lemma 2.1. For p and q ∈ H,σl(pI + qA) ={p + qt: t ∈ σl(A)}, where I is the identity matrix. Right spectrum does not enjoy Lemma 2.1. On the other hand, right spectrum ∗ is unitarily invariant, i.e. σr (U AU) = σr (A),whereU is a unitary matrix, but left spectrum is not unitarily invariant. Lemma 2.2. If A is a (upper or lower) triangular matrix, then σl(A) is the set of distinct diagonal elements of A. The right spectrum of a triangular matrix is the union of distinct similarity classes of its diagonal elements. Next we focus on 2 × 2matrix. Theorem 2.3. Let ab A = . cd (a) If bc = 0, then σl(A) ={a,d}. − − (b) If bc= / 0, then σl(A) ={a + bλ: λ2 + b 1(a − d)λ − b 1c = 0}. Proof. (a) A is a triangular matrix, and the result follows from Lemma 2.2. (b) Using Lemma 2.1, we have σ (A) = σ aI + b 01 l l b−1cb−1(d − a) = a + bσ 01. l b−1cb−1(d − a) Note that λ ∈ σ 01 l b−1cb−1(d − a) x if and only if there exists nonzero y such that x x 01 = λ b−1cb−1(d − a) y y x if and only if there exists nonzero y such that 108 L. Huang, W. So / Linear Algebra and its Applications 323 (2001) 105–116 y = λx, b−1cx + b−1(d − a)y = λy if and only if λ2 + b−1(a − d)λ − b−1c = 0. Hence the computation of the left spectrum of a 2 × 2 matrix is reduced to solving a quaternionic quadratic equation, whose solutions can be obtained by the quatern- ionic quadratic formulas in [4]. For the sake of completeness, we include this result as follows. Theorem 2.4. The solutions of the quadratic equation x2 + bx + c = 0 can be ob- tained by formulas according to the following cases: 1. If b, c, ∈ R and b2 < 4c, then x = 1 (−b + β + γ + δ ) 2 i j k for all β,γ,δ ∈ R with β2 + γ 2 + δ2 = 4c − b2. 2. If b, c, ∈ R and b2 > 4c, then √ −b b2 − c x = 4 . 2 3. If b ∈ R and c 6∈ R, then −b ρ c c c x = ∓ 1 i ∓ 2 j ∓ 3 k, 2 2 ρ ρ ρ where c = c0 + c1i + c2j + c3k,ci ∈ R, and v u q u 2 2 2 2 2 2 tb − 4c0 + (b − 4c0) + 16(c + c + c ) ρ = 1 2 3 . 2 4. If b 6∈ R, then − b x = Re − (b0 + T)−1(c0 − N), 2 where 0 0 Re b Re b b = b − Re b = Im b, c = c − b − , 2 2 and (T , N) is chosen√ as (a) T = 0,N= (B B2 − 4E)/2 provided that D = 0,B2 > 4E. p √ √ (b) T = 2 E − B, N = E provided that D = 0,B2 < 4E. √ (c) T = z, N = (T 3 + BT + D)/2T provided that D/= 0 where z is the unique positive root of the real polynomial z3 + 2Bz2 + (B2 − 4E)z − D2, where B =|b0|2 + 2Rec0,E=|c0|2, and D = 2Reb¯0c0. L. Huang, W. So / Linear Algebra and its Applications 323 (2001) 105–116 109 Example 2.5. Let " # 01+ i A = . 1 − i 0 Then n o 2 σl(A)= (1 + i)λ: λ + i = 0 ( √ ) 2 = (1 + i)λ: λ = (1 − i) 2 n√ √ o = 2, − 2 . Example 2.6. Let A = 0 i . j 1 Then n o 2 σl(A)= iλ: λ + iλ + k = 0 1 1 = iλ: λ = (1 − i − j − k), (−1 − i − j + k) 2 2 1 1 = (1 + i + j − k), (1 − i − j − k) . 2 2 Example 2.7. Let A = 2 i . −i 2 Then n o 2 σl(A)= 2 + iλ: λ + 1 = 0 1 = 2 − iλ: λ = (βi + γ j + δk), β2 + γ 2 + δ2 = 4 n 2 o = 2 − β − δj + γ k: β2 + γ 2 + δ2 = 1 .