A Bruhat decomposition for subgroups containing the group of diagonal matrices. II

This paper is a continuation of RZhMat 1981, 7A438. Suppose R is a commutative ring generated by its group of units R^* and there exist such that. Suppose also that is the Jacobson radical of R, and B() is a subgroup of GL(n,R) consisting of the matrices a=(a_ij) such that a_ij for i>j. If a matrix aB() is represented in the form a=udv, where u is upper unitriangular, d is diagonal, and v is lower unitriangular, then u,vD,aDa^–1, where D=D(n,R) is the group of diagonal matrices. In particular, D is abnormal in B()Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 114, pp. 50–61, 1982.In conclusion, the author would like to thank Professor Z. I. Borevich for many useful discussions. The present paper (like [5, 12] and, in part [6, 7]) was mainly written in 1979–1980, while the author was at Wroclaw University. The author would like to thank his Wroclaw colleagues and, primarily, his adviser, Professor W. Narkiewicz, for creating ideal conditions under which to work.     

Normalizer of the subgroup of diagonal matrices in the general linear group over a ring

The normalizer of the diagonal subgroup in the general linear group over certain classes of rings is determined. Sufficient conditions are given under which this normalizer coincides with the group of monomial matrices.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 132, pp. 114–118, 1983.     

Subgroups of the orthogonal group that contain a group of quasidiagonal matrices

In the group of real orthogonal matrices we study the lattice of subgroups containing a group of quasidiagonal orthogonal matrices of fixed type with at most one first-order block.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 94, pp. 73–80, 1979.     

The subgroups of the special linear group over a skew field that contain the group of diagonal matrices

For any (noncommutative) skew field T, the lattice of subgroups of the special linear group Λ=SL(n,T) that contain the subgroup Δ=SD(n,T) of diagonal matrices (with Dieudonné determinants equal to 1) is studied. It is established that for any subgroup H, Δ≤H≤Λ, there exists a uniquely determined unital net σ such that Λ(σ)≤H≤N(σ), where Λ(σ) is the net subgroup associated with the net σ and N(σ) is its normalizer in Λ. Bibliography: 11 titles. Published inZapiski Nauchnykh Seminarov POMI, Vol. 211, 1994, pp. 91–103. Translated by Bui Xuan Hai.     

Some results on symplectic matrices

The principal results are that if A is an integral matrix such that AA^T is symplectic then A = CQ, where Q is a permutation matrix and C is symplectic; and that if A is a hermitian positive definite matrix which is symplectic, and B is the unique hermitian positive definite pth.root of A, where p is a positive integer, then B is also symplectic.     

Some results on symplectic matrices

The principal results are that if A is an integral matrix such that AA^T is symplectic then A = CQ, where Q is a permutation matrix and C is symplectic; and that if A is a hermitian positive definite matrix which is symplectic, and B is the unique hermitian positive definite pth.root of A, where p is a positive integer, then B is also symplectic.     

Linear stability of some symplectic matrices

This paper is devoted to a study on the linear stability of some basic normal forms of symplectic matrices. Some necessary and sufficient conditions are given.     

Lyapunov diagonal semistability of acyclic matrices

It is shown that an acyclic matrix is Lyapunov diagonally semistable if and only if the matrix has the weak principal submatrix rank property. This result completes the solution of the problem of characterizing the various types of matrix stability for acyclic matrices. Also, those acyclic matrices which have a unique Lyapunov scaling factor are characterized.     

Subgroups of the Clifford group

LetF be a commutative field. We introduce in the setA=(F\{0})×F the following binary operation ·:A×A A:

Counterexamples concerning the diagonal elements of normal matrices

An infinite class of conditions known to be satisfied by the diagonal elements of a normal matrix with prescribed spectrum is shown to be independent of other known conditions satisfied by the diagonal elements but nevertheless insufficient to characterize the diagonal.