Making use of an elementary fact on invariant subspace and determinant of a linear map and the method of algebraic identities, we obtain a factorization formula for a general characteristic polynomial of a matrix. This answers a question posed in [A. Deng, I. Sato, Y. Wu, Characteristic polynomials of ramified uniform covering digraphs, European Journal of Combinatorics 28 (2007), 1099–1114]. T...

A sampling theorem for regular sampling in shift invariant subspaces is established. The sufficient-necessary condition for which it holds is found. Then, the theorem is modified to the shift sampling in shiftinvariant subspaces by using the Zak transform. Finally, some examples are presented to show the generality of the theorem.

Cross-modal learning tries to find various types of heterogeneous data (e.g., image) from a given query (e.g., text). Most cross-modal algorithms heavily rely on semantic labels and benefit from a semantic-preserving aggregation of pairs of heterogeneous data. However, the semantic labels are not readily obtained in many real-world applications. This paper studies the aggregation of these pairs...

It is shown that if a paranormal contraction T has no nontrivial invariant subspace, then it is a proper contraction. Moreover, the nonnegative operator Q = T 2∗T 2 − 2T ∗T + I also is a proper contraction. If a quasihyponormal contraction has no nontrivial invariant subspace then, in addition, its defect operator D is a proper contraction and its itself-commutator is a trace-class strict contr...

This paper describes two methods for computing the invariant subspace of a matrix. The rst method involves using transformations to interchange the eigenvalues. The matrix is assumed to be in Schur form and transformations are applied to interchange neighboring blocks. The blocks can be either one by one or two by two. The second method involves the construction of an invariant subspace by a di...

In this work, we develop a geometric method for solving the problem of H2-optimal rejection of disturbance inputs in continuous-time linear systems without feedthrough terms from the control input and the disturbance input to the controlled output and the measured output. A necessary and sufficient condition for the solvability of the problem is expressed in terms of a pair of subspaces, a cont...

A geometric approach for systems represented by a singular 2D Fornasini-Marchesini model is developed by introducing suitable notions of invariant subspace and controlled invariant subspace of the state space. The first notion is shown to be usefull in characterizing the set of compatible boundary conditions and in studying the existence and uniqueness of solutions to the state space equation o...

This paper presents the Cholesky factor–alternating direction implicit (CF–ADI) algorithm, which generates a low rank approximation to the solution X of the Lyapunov equation AX + XAT = −BBT . The coefficient matrix A is assumed to be large, and the rank of the righthand side −BBT is assumed to be much smaller than the size of A. The CF–ADI algorithm requires only matrix-vector products and mat...

The Rayleigh-Ritz method finds the stationary values, called Ritz values, of the Rayleigh quotient on a given trial subspace as approximations to eigenvalues of a Hermitian operator A. If the trial subspace is A-invariant, the Ritz values are exactly some of the eigenvalues of A. Given two subspaces X and Y of the same finite dimension, such that X is A-invariant, the absolute changes in the Ri...

We define center manifold as usual as an invariant manifold, tangent to the invariant subspace of the linearization of the mapping defining a continuous dynamical system, but the center subspace that we consider is associated with eigenvalues with small but not necessarily zero real parts. We prove existence and smoothness of such center manifold assuming that certain inequalities between the c...