The Sum of Two Matrices

The sum of two matrices is defined as the matrix whose elements are the sum of the elements of the two original matrices. Thus if we add matrix with elements Oij to matrix with elements, we get a matrix C whose elements c j are given by Eq, (15.2). 

The sum of two matrices is obtained by adding the corresponding elements of the two matrices. For example, given... 

Let s now look at the density matrix as it evolves for the pulse sequences that we outlined in Section 11.5. We have illustrated INEPT and related pulse sequences by the two-spin system in which I = H and S = 13C.The equilibrium density matrix can be considered as the sum of two matrices ... 

A basic condition for the sum of two matrices is that the two matrices have the same dimension. 

Matrices and mathematical operators have some things in common. There is a well-defined matrix algebra in which matrices are operated on and this matrix algebra is similar to operator algebra. Two matrices are equal to each other if and only if both have the same number of rows and the same number of columns and if every element of one is equal to the corresponding element of the other. The sum of two matrices is defined by... 

In fault diagnosis and in robust control of systems with parameter uncertainties (1.2a, 1.2b) is commonly used in a form in which the matrices are decomposed into the sum of two matrices [9-11]. One of them only depends on the nominal system parameters 0 , the other one accounts for parametric faults or for model parameter uncertainties A and may be constant or time varying. 

Then let us take into account the form of the matrices V. It represents an off-diagonal matrix block having nonvanishing matrix elements only if one of the vectors (bra) belongs to the subset of the occupied MOs and another (ket) to the subset of the vacant MOs. Then the only relevant part of the perturbation matrix W is the sum of two similar conjugate off-diagonal blocks ... 

The on-shell T-matrix is related to the observables that are measured in experiment. Thus, potentials which fit the same NN scattering data produce the same on-shell T-matrices. However, this does not imply that the potentials are the same. As seen in Eqs. (1) and (2), the T-matrix is the sum of two terms, the Born term and an integral term. When this sum is the same, the individual terms may still be quite different. 

Another matrix approach uses the split operator method."" This makes use of the approximate form of equation (72) for the exponential function of the sum of two general matrices A and B ... 

L i I F theory also uses two density matrices, the full density matrix being the sum of these two ... 

From our previous chapter defining the elementary matrix operations, we recall the operation for multiplying two matrices the i, j element of the result matrix (where i and j represent the row and the column of an element in the matrix respectively) is the sum of cross-products of the /th row of the first matrix and the y th column of the second matrix (this is the reason that the order of multiplying matrices depends upon the order of appearance of the matrices - if the indicated ith row and y th column do not have the same number of elements, the matrices cannot be multiplied). 

Thus for each mode a factorization (decomposition into scores and loadings) is done, expressed by three matrices and a three-way core-array G. The matrices A, B, C, and G are computed by minimizing the sum of squared errors the optimum number of factors, P, Q, R, can be estimated by cross validation. In a similar manner the Tucker2 model can be defined which reduces only two of the three modes, as well as the Tuckerl model which reduces only one of the three modes. 

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