In this paper, we determine the normal forms of idempotent matrices for similarity over finite local rings Z/p^kZ, from which we construct a Cartesian authentication code and compute its size parameters and the probab...In this paper, we determine the normal forms of idempotent matrices for similarity over finite local rings Z/p^kZ, from which we construct a Cartesian authentication code and compute its size parameters and the probabilities of successful impersonation and substitution attack under the hypothesis that the cecoding rules are chosen according to a uniform probability distribution.展开更多
Some rank equalities are established for anti-involutory matrices. In particular, we get the formulas for the rank of the difference, the sum and the commutator of anti-involutory matrices.
Let R be an abelian ring (all idempotents of R lie in the center of R), and A be an idempotent matrix over R. The following statements are proved: (a). A is equivalent to a diagonal matrix if and only if A is similar ...Let R be an abelian ring (all idempotents of R lie in the center of R), and A be an idempotent matrix over R. The following statements are proved: (a). A is equivalent to a diagonal matrix if and only if A is similar to a diagonal matrix. (b). If R is an APT (abelian projectively trivial) ring, then A can be uniquely diagonalized as diag{el, ..., en} and ei divides ei+1. (c). R is an APT ring if and only if R/I is an APT ring, where I is a nilpotent ideal of R. By (a), we prove that a separative abelian regular ring is an APT ring.展开更多
The paper researches the rank of combinations a PA＋bAQ-cPAQ of two idempotent matrices P and Q.Using the properties of the idempotent matrix and elementary block matrix operation,we get some rank equalities for combi...The paper researches the rank of combinations a PA＋bAQ-cPAQ of two idempotent matrices P and Q.Using the properties of the idempotent matrix and elementary block matrix operation,we get some rank equalities for combinations a PA＋bAQ-cPAQ of two idempotent matrices P and Q.These rank equalities generalize the results of Koliha J J,Rakoevi V and Tian Y,and give some applications of the rank equalities.展开更多
Let R be an associative unital ring and not necessarily commutative.We analyze conditions under which every n×n matrix A over R is expressible as a sum A=E1+…+Es+N of(commuting)idempotent matrices Ei and a nilpo...Let R be an associative unital ring and not necessarily commutative.We analyze conditions under which every n×n matrix A over R is expressible as a sum A=E1+…+Es+N of(commuting)idempotent matrices Ei and a nilpotent matrix N.展开更多
Using a limit process, it is proved in this paper that the adjoint matrix of an idempotent matrix is idempotent and the adjoint matrix of a nilpotent matrix is also nilpotent. The results are richer than that in [1].
基金Foundation item:The Key Project(03060)of Chinese Ministry of Education.
文摘In this paper, we determine the normal forms of idempotent matrices for similarity over finite local rings Z/p^kZ, from which we construct a Cartesian authentication code and compute its size parameters and the probabilities of successful impersonation and substitution attack under the hypothesis that the cecoding rules are chosen according to a uniform probability distribution.
文摘Some rank equalities are established for anti-involutory matrices. In particular, we get the formulas for the rank of the difference, the sum and the commutator of anti-involutory matrices.
文摘Let R be an abelian ring (all idempotents of R lie in the center of R), and A be an idempotent matrix over R. The following statements are proved: (a). A is equivalent to a diagonal matrix if and only if A is similar to a diagonal matrix. (b). If R is an APT (abelian projectively trivial) ring, then A can be uniquely diagonalized as diag{el, ..., en} and ei divides ei+1. (c). R is an APT ring if and only if R/I is an APT ring, where I is a nilpotent ideal of R. By (a), we prove that a separative abelian regular ring is an APT ring.
文摘The paper researches the rank of combinations a PA＋bAQ-cPAQ of two idempotent matrices P and Q.Using the properties of the idempotent matrix and elementary block matrix operation,we get some rank equalities for combinations a PA＋bAQ-cPAQ of two idempotent matrices P and Q.These rank equalities generalize the results of Koliha J J,Rakoevi V and Tian Y,and give some applications of the rank equalities.
基金supported by Ministry of Educations,Science and Technological Development of Republic of Serbia Project#174032.
文摘Let R be an associative unital ring and not necessarily commutative.We analyze conditions under which every n×n matrix A over R is expressible as a sum A=E1+…+Es+N of(commuting)idempotent matrices Ei and a nilpotent matrix N.
文摘Using a limit process, it is proved in this paper that the adjoint matrix of an idempotent matrix is idempotent and the adjoint matrix of a nilpotent matrix is also nilpotent. The results are richer than that in [1].
