Proper subspace

A vector space Lm is said to be a subspace of another vector space Ln if every vector of Lm is also contained in Ln. Lm is called a proper subspace of Ln if the vectors of Lm do not exhaust the space Ln. 

The vector space Ln, which is invariant under G, may contain a proper subspace which is also invariant under G. In such a case, Lm is an invariant subspace of Ln under G, and the space Ln is said to be reducible under G. 

If the Lm and Ln m subspaces contain further invariant (proper) subspaces within them the process of reduction can be carried on until no further unitary transformation can be found to further reduce the matrices of the representation. The final form of the matrices of the representation T may... 

The goal of this section is to find useful spanning subspaces of C[— 1, 1] and 2(52) Recall from Definition 3.7 that a subspace spans if the perpendicular subspace is trivial. In a finite-dimensional space V, there are no proper spanning subspaces any subspace that spans must have the same dimension as V and hence is equal to V. However, for an infinite-dimensional complex scalar product space the situation is more complicated. There are often proper subspaces that span. We will see that polynomials span both C[—l, 1] andL2(5 2) in Propositions 3.8 and 3.9, respectively. In the process, we will appeal to the Stone-Weierstrass theorem (Theorem 3.2) without giving its proof. 

Note that if desired, the Cartan subalgebra T may be interpreted as a proper subspace of transformations adp which corresponds to zero eigenvalue. The multiplicity of zero eigenvalue is equal to r, that is, to the rank of the algebra G of the dimension of the Cartan subalgebra (Fig. 14). 

The formulae obtained for K by CGM (and [3]) and [2], reduce to the proper result for the case where N = 1. Indeed, this is because, in such a case, there is only one possible orientation of the basis vector in the one-dimensional subspace (the subspace being fixed), and its phase is physically meaningless. 

Since Lm is invariant under G, any operator A G transforms each vector >n Lm into another vector in Lm. Hence, the operation AM results in a matrix of the same form as T(A). It should be clear that the two sets of matrices I) 1) and D > give two new representations of dimensions m and n — m respectively for the group G. For there exists a set of basis vectors l, n] for rX2 The representation T is said to be reducible. It follows that the reducibility of a representation is linked to the existence of a proper invariant subspace in the full space. Only the subspace of the first m components is... 

Theorem 4. The proper reactions over a set of species sTs lie in a subspace ofi8 of dimensions (s - t ), where t is the dimension of the subspace of 91t within which the set s/s lies. 

Proof. Consider the dual vector space 93s- This is isomorphic to an S dimensional vector space, 5ls, in which 2IT is embedded. Let b be the subspace of 93s which is isomorphic to the subspace of Us within which the set s/s lies. By Theorem 2 the dimension of b is t. Now the proper reactions of 93s are the subspace b of annihilators of b. By a well-known theorem on dual vector spaces [d], b must have dimension (s - t ). ... 

It has already been mentioned that the equation (116) of motion can be presented in a symmetric form provided that the basis sets of the individual subspaces of the composite space are properly normalized. Namely, if we substitute ... 

The point is that the vectors k 4 satisfying the unperturbed Schrodinger equation, if used to expand 44 make the right hand side disappear and the equation becomes a uniform one. The only thing we can do is to use it to determine the proper expansion coefficients of the zeroth order wave function b p in terms of the degenerate subspace as well as the first order energy. (The first order wave function is usually not calculated/considered in the degenerate case.)... 

The eigenvalue problem of the Hamiltonian operator (1) is defined in an infinite-dimensional Hilbert space Q and may be solved directly only for very few simple models. In order to find its bound-state solutions with energies not too distant from the ground-state it is reduced to the corresponding eigenvalue problem of a matrix representing H in a properly constructed finite-dimensional model space, a subspace of Q. Usually the model space is chosen to be spanned by TV-electron antisymmetrized and spin-adapted products of orthonormal spinorbitals. In such a case it is known as the full configuration interaction (FCI) space [8, 15]. The model space Hk N, K, S, M) may be defined as the antisymmetric part of the TV-fold tensorial product of a one-electron space... 

After a brief summary of the molecular and MO-communication systems and their entropy/information descriptors in OCT (Section 2) the mutually decoupled, localized chemical bonds in simple hydrides will be qualitatively examined in Section 3, in order to establish the input probability requirements, which properly account for the nonbonding status of the lone-pair electrons and the mutually decoupled (noncommunicating, closed) character of these localized a bonds. It will be argued that each such subsystem defines the separate (externally closed) communication channel, which requires the individual, unity-normalized probability distribution of the input signal. This calls for the variable-input revision of the original and fixed-input formulation of OCT, which will be presented in Section 4. This extension will be shown to be capable of the continuous description of the orbital(s) decoupling limit, when AO subspace does not mix with (exhibit no communications with) the remaining basis functions. 

There is an arbitrariness in the choice of dimensions ttx of the geminal orbital subspaces. However, one can argue that for the valence geminals the proper choice is tix = 2 as in the GVB/PP scheme. In this case there is one orbital for each electron. One is then guaranteed a qualitatively correct picture of the disruption of an electron-pair bond. 

The golden-rule rate expressions obtained and discussed above are very useful for many processes that involve transitions between individual levels coupled to boson fields, however there are important problems whose proper description requires going beyond this simple but powerful treatment. For example, an important attribute of this formalism is that it focuses on the rate of a given process rather than on its full time evolution. Consequently, a prerequisite for the success of this approach is that the process will indeed be dominated by a single rate. In the model of Figure 12.3, after the molecule is excited to a higher vibrational level of the electronic state 2 the relaxation back into electronic state 1 is characterized by the single rate (12.34) only provided that thermal relaxation within the vibrational subspace in electronic state 2 is faster than the 2 1 electronic transition. This is... 

The generalized valence bond (GVB) method developed by Gaddard and coworkers does not employ a full Cl in the valence shell, but includes only the configurations needed to describe proper dissociation of a chemical bond. As will be demonstrated later, such a wavefunction represents a restricted form of the CASSCF wavefunction, where the active subspace is partitioned into subsets with a fixed number of electrons occupying the orbitals in each subset (acutally the GVB function is further restricted by allowing only specific spin couplings within each subset). 

From the above result, it could be inferred exactly that such irreducible subspaces of the state space establish the proper mathematical domain of the classical physical field quantities. In fact, the demonstration was undertaken by using a relativistic electromagnetic field tensor, F,y, and its antisymmetric property ... 

Equation (7.C.2) is fundamental. It tells us that a rotation of a function in the Ith subspace of Hilbert space produces another function which still lies entirely in the /th subspace. We say, therefore, that the 1th subspace forms a 21 + 1 dimensional invariant subspace corresponding to the group of proper rotations. 

For degenerate ground-states, each potential V e F leads to a subspace of wavefunctions Py Now, since one potential leads to more than one ground-state wavefunction, C as defined previously is no longer a map. However, if V and V lead to subspaees Ey and Py., and differ by more than a constant, then the inverse map C P- F, where T is a union of the subspaces Ey, is a proper map. Certainly ground-state wavefunctions from the subspaces Ey and... 

The modified direct inversion in the iterative subspace (MDIIS) method combines the simplicity and relatively small memory usage of an iterational approach with the efficiency of a direct method. It comprises two stages minimization of the residual linearly approximated with last successive iterative vectors used as a current basis, and then update of the basis with the minimized approximate residual by a properly scaled parameter. 

For microfluidic systems with strong nonlinear effects, proper orthogonal decomposition (POD)-based MOR (also called Karhunen-Loeve decomposition) is employed to locate the low-dimensional subspace for modeling. POD is... 

Proper orthogonal decomposititMi is a technique that extracts the orthogonal basis function spanning the reduced subspace using an ensemble of data from experiments or numerical simulation of the original full systems. 

In choosing a basis, one can search for an optimum choice that gives successively the highest overlap of wave functions. Thus, let y(r) 5 and tt(r) e 5 and choose the maximum of l(xln)l with respect to both subspaces moreover, let lx,), Iq,) be the minimizing orbitals under the normalization conditions. By a proper choice of phases so that (xlq) is real, the following can be obtained ... 

The molecules are assumed to be trapped with a separation Az ry = (2nf /y) /, where the dipole-dipole interaction is d /fy = y/2. In this regime the rotation of the molecules is strongly coupled to the spin and the excited states are described by Hunds case (c) states in analogy to the dipole-dipole coupled excited electronic states of two atoms with fine structure. The ground states are essentially spin-independent. In the subspace of one rotational quantum (Hi -b H2 = 1), there are 24 eigenstates of Hm which are linear superpositions of two electron spin states and properly symmetrized rotational states of the two molecules. There are several symmetries that reduce Hin to block diagonal form. First, Hm, conserves the quantum number Y = M / + Ms where Mff = + Mh/2 Ms = Ms + Ms2 are the total rotational and spin projections... 

The proof is obvious P consists of complex structures J which have no proper invariant subspaces in Pq = Hj 0 Q. 

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