Orthogonal similarity via transversal intersection of manifolds

Let A be an n n real matrix.We consider a property of A called the orthogonal similarity-transversality property (OSTP) that is equivalent to the condition that the smooth manifold consisting of the real matrices orthogonally similar to A and the smooth manifold Q(sgn(A)) (consisting of all real matrices having the same sign pattern as A), both considered as embedded smooth submanifolds of R nn ,intersect transversally at A. More specifically, with S = [s i j ] being the n n generic skew-symmetric matrix whose strictly lower (or upper) triangular entries are regarded as independent free variables, we say that A has the OSTP if the Jacobian matrix of the entries of AS -SA at the zero entry positions of A with respect to the strictly lower (or upper) triangular entries of S has full row rank.We also formulate a property called OSTP2 that is equivalent to the OSTP: a square real matrix A satisfies the OSTP2 if X = 0 is the only matrix such that A X = 0 and AX T -X T A is symmetric.We show that if a matrix A has the OSTP, then every superpattern of the sign pattern sgn(A) allows a matrix orthogonally similar to A, and every matrix sufficiently close to A also has the OSTP.This approach provides a theoretical foundation for constructing matrices orthogonally similar to a given matrix while the entries have certain desired signs or zero-nonzero restrictions.In particular, several necessary conditions for a matrix to have the OSTP are given, and several important classes of zero-nonzero patterns and sign patterns that require or allow the OSTP are identified.Examples illustrating some applications (such as constructions of sign patterns that allow normality or orthogonality) are provided.Several problems are raised.