| This thesis is devoted to algebraic aspects of invertibility of infinite matrices. Based on discussion about infinite matrices over rings, a survey on invertibility of infinite matrices, focused on the recent research in China, is given. We summarize the sufficient and necessary conditions for an upper triangular matrix to be invertible. Proofs of some results are presented.Let R be a ring, M(R) the set of infinite matrices over R, RF(R) the ring of row-finite matrices, CF(R) the ring of column-finite matrices and RCF ( R) = RF ( R)∩CF ( R). We consider M(R) as an R-R-bimodule.In§2, we first introduce product of infinite matrices, discuss the associative law, and prove that M(R) is an RF(R)-CF(R)-bimodule isomorphic to the bimodule HomR(R(N), RN). This result covers discussion on associative law in the literature.Definition 2.1 Let A=(aij) and B=(bij) be both infinite matrices over the ring R. If for any i and j there exist only a finite number of nonzero elements in the sequence ai1b1j, ai2b2j,......, then we say AB to be well-defined,and set ( )AB =∑∞k=1 ai k bkj.The definition is different from that used in analysis. When R is the real or complex number field, the analytic definition of product of infinite matrices A and B over R is ( )AB =∑∞k =1 ai k bkj if the series ∞k =1 ai k bkj∑is convergent for any i and j. In this case, the multiplication of infinite matrices in analysis is more generalized than that in algebra. However, the definition in analysis is not available for matrices over rings.Though the multiplication of infinite matrices satisfies the distributive law (Proposition 2.1), it does not satisfy the associative law, even for three identical matrices (Example 2.1). The square of a symmetric matrix is well defined if and only if it is row-finite and column-finite (Proposition 2.2). We prove that M(R) is an RF(R)-CF(R)-bimodule (Proposition 2.4), which covers discussion about associative law in the literature. We also prove the following theorem.Theorem 2.1 Let R have identity 1, and consider R(N) as an R-RF(R)-bimodule and RN as an R-CF(R)-bimodule. Then (1)HomR(R(N), RN)?M(R) as RF(R)-CF(R)-bimodules. (2) EndR(R(N)) is isomorphic to RF(R) as rings.In§3, we discuss invertibility of infinite matrices and present the sufficient and necessary conditions of invertibility proved by Shiqiang Wang and others.The study about invertibility of infinite matrices in China began with Shiqiang Wang in 1993. He introduced the notation of infinitely linear independence and proved the following theorem. Definition 3.1 Let A be a column-finite matrix over a field K with the rows r1,r2,r3,…. If a1r1+a2r2+a3r3+….≠0 for any a1,a2,a3,…in K, not all zero, then we say that r1,r2,r3,…are infinitely linearly independent. Theorem 3.1 Let A be a row-finite matrix over a field K.(1) A has a right inverse if and only if the rows of A are linearly independent over K;(2) A has a left inverse which is row-finite if and only if the columns of A are infinitely linearly independent;(3) If A has a right inverse and a left inverse which is row-finite, then B has the unique inverse and the unique right inverse;(4) Let A∈RCF(K). Then (i) A is invertible in RF(R) if and only if the rows are linearly independent and the columns are infinitely linearly independent; (ii) A is invertible in RCF(R) if and only if the rows and columns of A are infinitely linearly independent, respectively.The results above have many generalizations, and one over division rings is given by Guolong Chen as follows.Theorem 3.2 Let A be a row-finite matrix over a division ring K.(1) A has a right inverse if and only if the rows are left linearly independent over K;(2) A has a left inverse which is row-finite if and only if the columns of A are right infinitely linearly independent over K;(3) If A has a right inverse and a left inverse which is row-finite, then A has the unique inverse and the unique right inverse;(4) Let A∈RCF(K). Then A has a left (right) inverse if and only if the columns (rows) of A are infinitely linearly independent.Moreover, if A has both left inverse and right inverse, one of which is in RCF(K), then A has the unique inverse. Chenyi Zhang proves the following theorems.Theorem 3.4 Let A be a row-finite matrix over a division ring.Then the following statements are equivalent.(1) A has a left inverse which is row-finite;(2) For any infinite matrix,AD=0 implies that D=0;(3) The columns of A are left infinitely linearly independent.By means of elementary operations and elementary matrices, further sufficient and necessary conditions of invertibility are given. In§4,we are concerned with invertible upper triangular infinite matrices. For such matrices, Guolong Chen prove the following results.Theorem 4.1 Let A be an upper triangular infinite matrix with right linearly independent columns. Then A has a right inverse which is an upper triangular infinite matrix.Theorem 4.2 Let A be an upper triangular rcf matrix with left linearly independent rows and right linearly independent columns. If a left inverse of A is row-finite, then A has the unique two-sided inverse. Qingxiu Li proves the following theorem.Theorem 4.5 Let A be an upper triangular infinite matrix. Then A has a two-sided inverse which is upper triangular if and only if the columns of A are right linearly independent.In 2004, Xianhua Guo studied the infinite matrices P = ??? B0A0??? , where B is a finite matrix and A is an upper triangular rcf matrix, and sufficient and necessary conditions for invertibility are obtained.
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