Element submatrix

I — 2vnoH)A is a matrix whose first column has only its first element nonzero. The same principle can now be applied to the submatrix that remains after removing the first row and first column of the transformed matrix, and so on until there results, finally, an upper triangular matrix. Notice that interchange of rows is not necessary. 

Now consider any principal submatrix of A. This is hermitian, and if a unitary transformation is applied to A so as to diagonalize this submatrix, then the sum of the squares of the diagonal dements of A is increased in the amount of the sum of the squares of the off-diagonal elements of the submatrix. In practice, the submatrices selected are of order 2. If the submatrix... 

Definition Submatrix. We choose a deliberate pixel from our scattering image. The pixel and its neighboring pixels are the submatrix. For the example we choose a submatrix size of 3 x 3 elements. There is scattering intensity in each pixel, e.g. 

The question is in the data from a real experiment, where many radionuclides are measured in many samples collected under a wide variety of conditions, what is the least number of classes of chemical behavior that will describe the observed results to the desired precision Or, in mathematical terms, what is the rank of the matrix A, and what nuclides should be selected to make up the submatrix a Finally, can any physical significance be attached to the combination of coefficients making up the elements of K, and can these elements of K or quantities thus derived be carried over from one event to the next ... 

The character of the matrix corresponding to the operation ah may be found without writing out any part of the complete matrix itself. We note that the operation ah does not shift any vectors from one atom to another. Hence no set of vectors may be summarily ignored. We note further that each set of vectors will be affected by ah exactly the same way. Thus whatever contribution to the character is made by one of the four sets may simply be multiplied by 4 in order to get the total value of the character. In any one set, ah transforms the X and Y vectors into themselves and the Z vector into its own negative. Thus the submatrix for this set of vectors will be diagonal with the elements 1,1, and -1 and hence a character of 1. The character of the entire matrix corresponding to the operation ah is thus 4. 

A set of spherical functions C (m = 0,+l,+2,...,+/) composes the so-called irreducible tensor. In fact, we shall not need their explicit expressions, only their one-electron reduced matrix (submatrix) elements... 

One-electron submatrix elements of the spherical functions operator occur in the expressions of any matrix element of a two-electron energy operator and the electron transition operators (except the magnetic dipole radiation), that is why we present in Table 5.1 their numerical values for the most practically needed cases /, / < 6. 

This submatrix element has the following fairly simple algebraic expression ... 

This submatrix element is always positive, and it is non-zero only when l + k + / is even. It is in the following way connected with a special case of the Clebsch-Gordan coefficient ... 

Thus, utilizing the concept of irreducible tensorial sets, one is in a position to develop a new method of calculating matrix (submatrix) elements, alternative to the standard way described in many papers [9-11, 14, 18, 21-23]. Indeed, the submatrix element of the irreducible tensorial operator can be expressed in terms of a zero-rank double tensorial (scalar) product of the corresponding operators (for simplicity we omit additional quantum numbers a, a1) ... 

Let us present the main definitions of tensorial products and their matrix or reduced matrix (submatrix) elements, necessary to find the expressions for matrix elements of the operators, corresponding to physical quantities. The tensorial product of two irreducible tensors and is defined as follows ... 

The quantity denoted ( ) is called a reduced matrix (submatrix) element of operator T k). It does not depend on projection parameters m, m, q. Dependence of the matrix element considered on these projections is contained in one Clebsch-Gordan coefficient. Such dependence is one of the indicators of the exceptional role played by the Clebsch-Gordan coefficients in the theory of many-particle systems. Their definitions and main properties will be discussed in the next paragraph. 

The submatrix element of the tensorial product of two operators, acting on one and the same coordinate, may be calculated applying the formula... 

The last multiplier on the right side of (5.19) is the 97-coefficient, defined in Chapter 6. Such expressions for submatrix elements directly follow from (5.19) in the case when the operator acts only on coordinates with index 1 or 2 ... 

Unit tensors play, together with spherical functions, a very important role in theoretical atomic spectroscopy, particularly when dealing with the many-electron aspect of this problem. Unit tensorial operator uk is defined via its one-electron submatrix element [22]... 

Let us also recall that the one-electron submatrix element of spin angular momentum operator s1 and of scalar quantity s° (in h units) is given by... 

It is worthwhile to emphasize that all dependences of matrix element (5.42) on the structure of a shell are contained in the items in square brackets, whereas all pecularities of the operators themselves are contained in one-electron submatrix elements. The matrix element of the second term in (5.42) is equal, for any k, to... 

In the general case, there is no simple algebraic expression for the first term of (5.42), therefore, for its evaluation one must utilize the tables of numerical values of the submatrix elements of operators Uk. The most complete tables, covering also the case of operators Vkl, may be found in [87], For operators, also depending on spin variables, the analogue of formula (5.41) will have the form... 

Table 7.3 presents numerical values of submatrix elements (7.5) for... 

It is interesting to emphasize that submatrix elements (7.5) are proportional to the CFPs with two detached electrons, defined by (9.15), namely... 

Let us present in conclusion the expression for the submatrix element of scalar product (7.3), necessary while calculating relativistic matrix elements of the energy operator. It is as follows ... 

The most effective way to find the matrix elements of the operators of physical quantities for many-electron configurations is the method of CFP. Their numerical values are generally tabulated. The methods of second-quantization and quasispin yield algebraic expressions for CFP, and hence for the matrix elements of the operators assigned to the physical quantities. These methods make it possible to establish the relationship between CFP and the submatrix elements of irreducible tensorial operators, and also to find new recurrence relations for each of the above-mentioned characteristics with respect to the seniority quantum number. The application of the Wigner-Eckart theorem in quasispin space enables new recurrence relations to be obtained for various quantities of the theory relative to the number of electrons in the configuration. 

If, according to [14], we introduce unit tensorial operator w kK, the submatrix element of which is... 

This can conveniently be done after the final expressions for the operators of various physical quantities have been derived, since it is for operators Uk and Vkl that detailed tables [87, 97] of their submatrix elements are available. 

This expression is the second-quantization form of an arbitrary two-electron operator with tensorial structure of the kind (14.57). Examination of this formula enables us to work out expressions for operators that correspond to specific physical quantities. Many of these operators possess a complex tensorial structure, and their two-electron submatrix elements have a rather cumbersome form [14]. Therefore, by way of example, we shall consider here only the most important of the two-electron operators - the operator of the energy of electrostatic interaction of electrons (the last term in (1.15)). If we take into account the tensorial structure of that operator and put its submatrix element into (14.61), we arrive at... 

The submatrix element that enters into this expression is, by (5.19), related to submatrix element (14.60) by... 

It is to be stressed that, although the two-electron submatrix elements in (14.63) and (14.65) are defined relative to non-antisymmetric wave functions, some constraints on the possible values of orbital and spin momenta of the two particles are imposed in an implicit form by second-quantization operators. Really, tensorial products (14.40) and (14.42), when the sum of ranks is odd, are zero. Thus, the appropriate terms in (14.63) and (14.65) then also vanish. 

The operator of the energy of electrostatic interaction of electrons in (14.65) is represented as a sum of second-quantization operators, and the appropriate submatrix element of each term is proportional to the energy of electrostatic interaction of a pair of equivalent electrons with orbital Lu and spin S12 angular momenta. The values of these submatrix elements are different for different pairing states, since, as follows from (14.66), the two-electron submatrix elements concerned are explicitly dependent on L12, and, hence, implicitly - on S12 (sum L12 + S12 is even). It is in this way that, in the second-quantization representation for the lN configuration, the dependence of the energy of electrostatic interaction on the angles between the particles shows up. This dependence violates the central field approximation. 

The second-quantization counterpart of this approach is the replacement (for the lN configuration) of operator (14.65) by some effective operator, whose two-particle submatrix elements are independent of characteristics L12, S12 of the pairing state of electrons. To this end, we introduce the submatrix element averaged over the number of various antisymmetric pairing states in shell, equal to (4/ + 2)(4/ + l)/2 ... 

The first factor under the summation sign takes care of the antisymmetry of the two-electron states of the shell. The submatrix elements are summed in accordance with the statistical weights of these states. Using (14.66), (6.25), (6.26) and (6.18), we sum the right side of (14.67) in the explicit form... 

Submatrix elements of creation and annihilation operators. Coefficients of fractional parentage... 

The submatrix elements of the creation operator, according to (5.15), can be given by the integral not depending on Ml, Ms, namely... 

---