Matrices Over Commutative Rings And Its Application To Partitioned Matrices With Commuting Blocks

This thesis concerns with diagonalization of matrices over a commutative ring R such that the free module R n is cohopfian, to generalize results in the literature about diagonalization of partitioned matrices with commuting blocks. We correct mistakes appeared in the literature.Let M_m(C) m denote the ring of m×m matrices over the complex number field. It is well known that any commuting subset of M_m(C) m is contained in a maximal commutative subring. For a maximal commutative subring P_m = P_m of M_m, let M_p,q P_m denote the set of p×q matrices over P_m , abbreviated ( )M r P_m whenever p = q = r. The matrices in M_p,q P_m is called matrices with commuting blocks..In 1971, Dennis et. al. introduced block eigenvalues for partitioned matrices. Vitória studies the case of matrices with commuting blocks and gives conditions of diagonalization of such matrices. Vitória also introduces a generalization of compound matrices---block compound matrices, which is compound matrices over P_m(C) . However several Chinese authors called every matrix over P_m(C) block compound matrix ambiguously. So we prefer to use the notion of matrices with commuting blocks to name the matrices over P_m(C) . Let ( )A∈M_n P_m and ( )Λ∈P_m, if AX = XΛfor some X∈M_n ,1P_m of full rank, thenΛis called a block eingenvalue of A and X is called the corresponding block eigenvector of A.The eigenvalues ofΛare eigenvalues of A. Conversely, a set of block eigenvalues of A is called a complete set if all eigenvalues of these block eigenvalues are just all eigenvalues of A.Vitória proves that for a complete set { }Λ1 ,…,Λn of block eigenvalues, if Vandermonde matrix is nonsingular, then the corresponding block eigenvetors X₁ , X₂, , X_n is block-linearly independent, and ( ) ( ) ( )1X₁ , X₂ , , X_n ? A X₁ , X₂ , , Xn = diagΛ1 ,Λ2, ,Λn.In 1999, Pang called the matrices over P_m(C) block compound matrices. He gives necessary and sufficient conditions of diagonalization of so-called block compound matrices. There is a gap in his paper and we fix it in this thesis.In§2, we generalize Cramer rule, invertibility and Vandermonde matrices to matrices over a commutative ring.Let R denote commutative ring with 1, and ( )M_n(R) denote the n×n matrix ring over R.Theorem 2.1 Let ( )A∈M_n(R). Then the columns of A is linearly independent if and only if the system of linear equations Ax = 0 has only the trivial solution.Theorem 2.3 Let ( )A∈M_n(R). Then the following conditions are equivalent:(1) A is invertible.(2) detA is invertible.(3) The columns of A is a free basis of R n.Theorem 2.4 Let r₁ , r₂ , , rn∈R. Then the Vandermonde matrix ( )V r₁ , r₂ , , rn is invertible if and only if ri ? rj,1≤j < i≤n,are all invertible.In§3 we generalize diagonalization of scalar matrices to matrices with commuting blocks.Definition 3.1 For A∈M_n(R) and d∈R, if there exists an independent elementξ∈Rⁿ, such that Aξ= dξ, we called d is an R ? eigenvalue (shortly, eigenvalue) andξa corresponding R ?eigenvector (shortly, eigenvector).Theorem 3.1 For A∈M_n (P_m), the block eigenvectors coincide with (P_m) ? eigenvalues.Theorem 3.2 Let A, P , D∈M_n( R), P invertible and D = diagλ₁ ,λ₂ , ,λ_n. Then P ?1 AP = D if and only if the columns of P are the corresponding eigenvectors, respectively.Theorem 3.3 Let A∈M_n(R). Then A is similar to a diagonal matrix if and only if R n has a free basis consisting of eigenvectors of A.Theorem 3.4 For ( )A∈M_nR with eigenvaluesλ₁ ,λ₂ , ,λ_s and corresponding eigenvectorsα₁ ,α₂, ,α_s∈Rn. Ifλi ?λj,1≤i < j≤s,are not zero divisors, thenα₁ ,α₂, ,α_s is linearly independent.A module is called cohopfian if its monomorphisms are all automorphisms.Artinian modules, especially finitely generated modules over ( )(P_m) are cohopfian.Theorem 3.5 For a position integer n , and a ring R , the following are equivalent:(1) Rⁿ is a cohopfian R ? module.(2) n linearly independent elements is a free basis. (3) matrices with linearly independent columns are invertible.Theorem 3.6 Let R is a ring such that R n is a cohopfian R ? module, and ( )A∈M nR,with eigenvaluesλ₁ ,λ₂ , ,λ_n . If the Vandermonde matrix ( )Vλ₁ ,λ₂ , ,λ_n is invertible, then A is similar to ( )diagλ₁ ,λ₂ , ,λ_n .In the literature, there exists an incorrect statement: for a complete setΛ₁ ,Λ₂, ,Λ_nof block eigenvalues of A, if A is block-similar to ( )diag (Λ₁ ,Λ₂, ,Λ_n), then VΛ₁ ,Λ₂, ,Λ_n is invertible.It is a routine matter to check that 1 with and is a counter example. Theorem 3.7 Let A∈M_n (P_m) with a complete set of block eigenvaluesΛ₁ ,Λ₂, ,Λ_n. If A has distinct eigenvalues, then V(Λ₁ ,Λ₂, ,Λ_n is invertible.