ϕSϕS-Orthogonal matrices

Linear Algebra and its Applications, 2011

Let S ∈ M 2n be skew-symmetric and nonsingular. For X ∈ M 2n , we show that the following are equivalent: (a) X has a φ S po-

Linear Algebra and its Applications, 2013

Let S ∈ S n be given. If Q is φ S orthogonal, then φ S (φ S (Q )) = φ S Q −1 = Q . If T is φ S symmetric, then φ S (φ S (T)) = φ S (T) = T. If R is φ S skew symmetric, then φ S (φ S (R)) = φ S (−R) = R. Proposition 2. Let S ∈ S n and let A ∈ M n (C) be given. The following are equivalent.

Linear Algebra and its Applications, 2009

It is known that if a matrix has a φ J polar decomposition, then it is of even rank. We provide necessary and sufficient conditions for a 2n-by-2n matrix of rank 2 to have a φ J polar decomposition.

Linear Algebra and its Applications, 2010

We present new results on the φ J polar decomposition of matrices.

Linear Algebra and its Applications, 2014

Linear Algebra and its Applications, 1982

In connection wilh the problem of finding the be~t projectio~ of k-dimensional spaces embedded in ,,-dimensional spaces Hennann Konig a,ked, Given mER and nEr\!, are there "X" matrices C={c,,), I,i=I, ... ,n, such that c,,=m for all i, IC'il=l lor ;"Pi, and C2={m2+n-l)l"~ Kilnig was e.\pecially intere,ted in symmetric C, and we find some families of malnee, ,ati,fying this condition. We also find SOme families of matrices satisfying the le.s~ re,;lrichve condition ee T = (m~ + n-1)1".

Journal of Combinatorial Theory, Series B, 1971

We define an n-type (1,-1) matrix N = Z + R of order n-2 (mod 4) to have R symmetric and RZ = (n-l)Z,. These matrices are analogous to skewtype matrices M = Z + W which have W skew-symmetric. lf n is the order of an n-type matrix, hl and h, the orders of Hadamard matrices, h the order of a skew-Hadamard matrix, and p7 = 1 (mod 4) is a prime power then we show there are: n-type matrices of orders p' + 1, (h-1)2 + 1, (n-1)2 + 1, (n-1)3 + 1; symmetric Hadamard matrices of orders 2n, 2n(n-l), 2pr(p7 + 1); Hadamard matrices of orders h,n, h,h,n(n-l), h,h,n(n-3) (this latter with n + 4 also the order of an n-type matrix); Hadamard matrices of orders 452, 612, 2452 and 3044, all "new." We also give existence conditions for many other classes of Hadamard matrices and another formulation for Goldberg's skew-Hadamard matrix of order (h-1)3 + 1.

1972

We study the conjecture: There exists a square (O,l,-l)-matrix W = W(w,k) of order w 8ati~fying = kI w for aZZ k = 0, 1, ..• , w when w = ° (mod 4).

arXiv: Rings and Algebras, 2020

This thesis examins a generalisation of polar decompositions to indefinite inner product spaces. The necessary general theory is studied and some general results are given. The main part of the thesis focuses on polar decompositions with commuting factors: First, a proof for a generalisation of the link between polar decomposition with commuting factors and normal matrices is given. Then, some properties of such decompositions are studied and it is shown that the commutativity of the factors only depends on the selfadjoint part. Eventually, polar decompositions with commuting factors are studied under similarity transformations that do not alter the structure of the space. For this purpose, normal forms are decomposed and analysed.

Journal of Algebra, 2007

Let F be an algebraically closed field of characteristic different from 2. Define the orthogonal group, On(F), as the group of n by n matrices X over F such that XX ′ = In, where X ′ is the transpose of X and In the identity matrix. We show that every nonsingular n by n skew-symmetric matrix over F is orthogonally similar to a bidiagonal skew-symmetric matrix. In the singular case one has to allow some 4-diagonal blocks as well. If further the characteristic is 0, we construct the normal form for the On(F)-similarity classes of skew-symmetric matrices. In this case, the known normal forms (as presented in the well known book by Gantmacher) are quite different. Finally we study some related varieties of matrices. We prove that the variety of normalized nilpotent n by n bidiagonal matrices for n = 2s + 1 is irreducible of dimension s. As a consequence the skew-symmetric nilpotent n by n bidiagonal matrices are shown to form a variety of pure dimension s.