CONJUGATE MATRICES

Conjugate points, triangular matrices, and Riccati equations

Let A be a real continuous nxn matrix on an interval T3 and let the n-vector x be a solution of the differential equation x = Ax on r. If [oc,g]er, g is called a conjugate point of a if the equation has a nontrivial solution vector x = (x1,...,xn) such that x1(a) = ... = xk(a) = x k+1(P) = ... = x (0) = 0 for some ke[l,n-l]. It is shown that the absence on (t..,t ) of a point conjugate to t, wi...

On nonsymmetric saddle point matrices that allow conjugate gradient iterations

Linear systems in saddle point form are often symmetric and highly indefinite. Indefiniteness, however, is a major challenge for iterative solvers such as Krylov subspace methods. It has been noted by several authors that a simple trick, namely negating the second block row of the saddle point system, leads to an equivalent linear system with a nonsymmetric coefficient matrix A whose spectrum i...

The conjugate gradient algorithm applied to quaternion - valued matrices

The well known conjugate gradient algorithm (cg-algorithm), introduced by Hestenes & Stiefel, [1952] intended for real, symmetric, positive definite matrices works as well for complex matrices and has the same typical convergence behavior. It will also work, not generally, but in many cases for hermitean, but not necessarily positive definite matrices. We shall show, that the same behavior is s...

E-Clean Matrices and Unit-Regular Matrices

Let $a, b, k,in K$ and $u, v in U(K)$. We show for any idempotent $ein K$, $(a 0|b 0)$ is e-clean iff $(a 0|u(vb + ka) 0)$ is e-clean and if $(a 0|b 0)$ is 0-clean, $(ua 0|u(vb + ka) 0)$ is too.

TR-2004014: A Homotopic/Factorization Process for Toeplitz-like Matrices with Newton's/Conjugate Gradient Stages