Some Researches On The Coneigenvalues Of Matrices

Let A = (a_ij)∈C^n×n. If there existλ∈C and x∈Cⁿ\{0} such thatλis said to be a coneigenvalue of A(in short c-eigenvalue), and x is said to be the coneigenvector of A corresponding toλ(in short c-eigenvector), where C(E) is the set of all complex(real) numbers, and C^n×n(R^n×n) denotes the set of all n×n complex(real) square matrices.Let A = (a_ij)∈C^n×n. If there existsλ∈C such thatλis a coneigenvalue of A, the nonnegative coneigenvalue |λ| equal to module ofλis said to be the basic coneigenvalue of A. Andλ_c⁺(A) denotes the set of all basic coneigenvalues of A.Let A, B∈C^n×n. If there exists a nonsingular matrix S∈C^n×n such that , we call that A is consistently similar to B (in short C-similar). If S can be taken as a unitary matrix, we call that A is U-consistently similar to B(in short unitary C-similar). Let A∈C^n×n. If there exists a nonsingular matrix S∈C^n×n such that S^-1AS|- is a upper triangular matrix, A is said to can be consistent triangularization(in short C-triangularization). If S can be taken such that S^-1AS|- is a diagonal matrix, A is said to can be consistent diagonalization(in short C-diagonalization). A is said to can be unitary C-triangularization and unitary C-diagonalization, if A is the simplified form we wanted by C-similarity of unitary matrix. In 1985, R.A.Horn and C.R.Johnson gave the concept of matrix coneigen-value and discussed some fundamental properties when they studied the theories of a complex symmetrical matrix. Although the coneigenvalues of a matrix are similar to general eigenvalues, there exists difference between them in essence. The coneigenvalues of a matrix play an important role in all kinds of canonical forms, different decomposition forms , C-similarity theory and C-diagonal theory under C-similarity. In addition, there are important applications for the coneigenvalues of a matrix in most computations of modern stochastic process and solving some physics problems by two-order linear partial differential equation. Y.P.Hong, R.A.Horn,S.Furtado and C.R. Johnson etc. further discussed the properties and applications of coneigenvalues respectively, and some important results were obtained.In this paper, we mainly study the equivalent representation and the relative properties of coneigenvalues, discuss the inclusion interval of coneigenvalues, and give some results of coneigenvalues for a special class of matrices. The main content contains three parts, i.e. the second chapter, the third chapter and the fourth chapter.In second chapter, we mainly study the equivalent representation and the relative properties of coneigenvalues, score coneigenvalues of matrix from different aspects, and analyze the difference and similarity between coneigenvalues and general eigenvalues. Meanwhile, we refer to the sufficient and necessary conditions of the matrix basic coneigenvalue set as a complete set. The main results are as follows.Theorem 2.1.2 Let A = (a_ij)∈C^n×n, thenλ∈λ_c(A) (?) there exists D= diag(e^iθ₁, e^iθ₂,…,e^iθ_n),θ_i∈R, i = 1,2,…, n, such thatλ∈λ(DA), where D satisfies DX = X|-, and X|- is the eigenvalue of DA with corresponding eigenvalueλ.Theorem 2.1.3 Let A = A_r + iA_i∈C^n×n, A_r, A_i∈R^n×n,λ=λ_r + iλ_i∈C\{0},λ_r,λ_i∈R, then whereTheorem 2.1.4 Let A∈C^n×n,λ≥0, then , whereTheorem 2.1.5 Let A = A_r + iA_i∈C^n×n, A_r, A_i∈R^n×n,0 <λ∈R, then , whereLet A = (a_ij)∈C^n×n. Ifλis k-repeated nonnegative eigenvalue of AA|-, we call thatλ^1/2 is k-repeated eigenvalue of A. If |λ_c⁺(A)| = n(multiple number is considered),λ_c⁺(A) is called a complete set.Theorem 2.2.1 Let A,B∈C^n×n. If A is C- simialr to B, we haveλ_c⁺(A) =λ_c⁺(B).Theorem 2.2.2 Let A = A_r + iA_i∈C^n×n, A_r, A_i∈R^n×n If A_r and A_i are exchangeable, and both of them are symmetric, thenλ_c⁺(A) is complete, andλ_c⁺(A) = {(σ_i²(A_r) +σ_i²(A_i))^1/2, i∈< n >}, whereσ_i(A_r) andσ_i(A_i) are the singular values of A_r and A_i respectively.Theorem 2.2.3 Let A = A_r+iA_i∈C^n×n, A_r, A_i∈R^n×n, then the following conditions are equivalent:1)λ_c⁺(A) is a complete set, and |λ_c⁺(A)| = n(multiple number is not considered).2)λ(AA|-) = {0≤μ₁ <μ₂ <…<μ_n}.3) there exist nonsingular matrix X∈C^n×n and∧= diag(μ_i)₁ⁿ, 0≤μ₁ <μ₂ <…<μ_n) such that AA|- = X∧X^-1.4) there exist nonsingular matrix S∈C^n×n and∧_A = diag(λ_i)₁ⁿ, 0≤λ₁ <λ₂ <…<λ_n, such that A = S∧_A(S|-)^-1.In the third chapter, the inclusion interval of coneigenvalues of a matrix is discussed: firstly, we give the inclusion interval of coneigenvalues of a classical Gerschgorin matrix, discuss several improved results, and then we get the inclusion interval of coneigenvalues corresponding to a compound matrix. The main results are as follows.Theorem 3.1.1 Let A = (a_ij)∈C^n×n, thenwhere G_i(A) = {z∈R⁺ : |z - |a_ii||≤r_i(A)}, i∈< n >, ,...