Matrix inner product

In polyensor form, the interaction energy between moments at site A and moments at site B is a matrix inner product ... 

Note that, as for the vertex-weighted distance matrix, the matrix inner product leads to an unsymmetrical matrix. 

This construction requires one matrix-vector multiplication with S and two inner products in each recursive step. Therefore, it is not necessary to store S explicitly as a matrix. The Lanczos process yields the approximation [21, 7, 12]... 

The projection of the X data vectors onto the first eigenvector produces the first latent variable or pseudomeasurement set, Zx. Of all possible directions, this eigenvector explains the greatest amount of variation in X. The second eigenvector explains the largest amount of variability after removal of the first effect, and so forth. The pseudomeasurements are called the scores, Z, and are computed as the inner products of the true measurements with the matrix of loadings, a ... 

The overlap integrals form the inner products of the linear space of the AOs 0j. Due to a confusion between the two roles of differentials, the matrix S is sometimes called the metric of the linear space. A metric m involving the 0i must satisfy m(0 , 03) < m(0i, 02) + m(02,03) and m(0i, 02) = 0 (i = 02), hence m(0i, 02) > 0. Clearly the overlap matrix satisfies none of these requirements. A genuine metric can be defined in terms of S the matrix M = Z - S satisfies the above axioms where Z is a matrix containing unity in every position. 

The quantity f g is called the inner product (or scalar product) of the column vectors f and g. The inner product is a generalization of the dot product (1.55) to vectors with an arbitrary number of complex components. Since a Hermitian operator satisfies (1.13), then (2.63) shows that for a Hermitian matrix A... 

The correct form of the inner product is often just mentioned in passing when complex rotation is discussed, and then usually only for a rotation of an originally real matrix representation of the Hamiltonian. A clearer understanding can be obtained by going back to matrix algebra where the form of the inner product is a direct consequence of the symmetry of the matrix. 

When the true intrinsic rank of a data matrix (the number of factors) is properly determined, the corresponding eigenvectors form an orthonormal set of basis vectors that span the space of the original data set. The coordinates of a vector a in an m-dimensional space (for example, a 1 x m mixture spectrum measured at m wavelengths) can be expressed in a new coordinate system defined by a set of orthonormal basis vectors (eigenvectors) in the lower-dimensional space. Figure 4.14 illustrates this concept. The projection of a onto the plane defined by the basis vectors x and y is given by a. To find the coordinates of any vector on a normalized basis vector, we simply form the inner product. The new vector a, therefore, has the coordinates a, = aTx and a2 = aTy in the two-dimensional plane defined by x and y. 

Mass transport is understood to mean the molecular diffusion in, out and through plastic materials like that shown schematically in Fig. 1-3. This figure represents most applications where there is a layer of plastic material separating an external environmental media from an inner product media. The product can be a sensitive medium with a complex chemical composition, e.g. food, that must be protected from external influences such as oxygen and contaminants. It can also be an aggressive chemical that must not escape into the surrounding environment. Because this plastic material barrier layer usually includes low molecular weight substances incorporated into the polymer matrix, there are many applications in which the transport of these substances into the product and environment must be minimized. 

We generally denote scalars by lowercase Greek letters (e.g., P), column vectors by boldface lowercase Roman letters (e.g., x), and matrices by capital italic Roman letters (e.g., H). A superscriptT denotes a vector or matrix transpose. Thus xT is a row vector, xTy is an inner product, and AT is the transpose of the matrix A. Unless stated otherwise, all vectors belong to R , the u-dimen-sional vector space. Components of a vector are typically written as italic letters with subscripts (e.g., xux2,.. . , ). The standard basis vectors in R" are the n vectors ei,e2,. . . , e , where e has the entry 1 in the th component and 0 in all others. Often, the associated vector norm is the standard Euclidean norm, j 2, defined as... 

Therefore, the first product (r) Du(r, w) in (15.241) above is a (matrix) multiplication of a row vector and a column vector, while the second product G is the tensorial inner product of this ordinary vector with the tensor G ". In particular, it is easy to show that... 

The dot or inner product of two vectors is a scalar quantity and is, in matrix notation. 

A row matrix of order m can be multiplied by a column matrix of order m in two different ways. The inner product is a scalar defined by... 

A general way of exploiting this sparseness in both the gradient and the Hessian construction involves the use of outer product algorithms to perform the matrix element assembly. In the case of the matrix multiplications required in the F matrix construction, this simply means that the innermost DO loop is over X in Eq. (260). (If t were the innermost DO loop, the result would be a series of dot products or an inner product algorithm.) When an outer product algorithm is used, the magnitude of the density matrix elements may be tested and the innermost DO loop is only performed for non-zero elements. (In the case of Hessian matrix construction, the test may occur outside of the two... 

Another approach to the C matrix construction is a CSF-driven approach proposed by Knowles et al.. With this approach, the density matrix elements dlgrs ars constructed for all combinations of orbital indices p, q, r and s, but for a fixed CSF labeled by n. Each column of the matrix C is constructed in the same way that the Fock matrix F is computed except that the arrays D" and d" are used instead of D and d. As with the F matrix construction described earlier, there are two choices for the ordering of the innermost DO loops. One choice results in an inner product assembly method while the other choice results in an outer product assembly method. The inner product choice, which does not allow the density matrix sparseness to be exploited, results in SDOT operations of length m or about m, depending on the integral storage scheme. The outer product choice, which does allow the density matrix sparseness to be exploited, has an effective vector length of n, the orbital basis dimension. However, like the second index-driven method described above, this may involve some extraneous effort associated with redundant orbital rotation variables in the active-active block of the C matrix. 

It is useful to compare these approaches when applied to a wavefunction expansion that results in a sparse density matrix. For example with a PPMC expansion, each d , with about possible unique elements, contains only about non-zero elements m(m -I- 2)/8 non-zero elements of the typ)e and mil non-zero elements of the type For m = 20 the matrix d" is only 0.29% non-zero. The inner product CSF-driven approach is clearly not suited for the sparse transition density matrix resulting from this type of wavefunction. The outer product CSF-driven approach does account for the density vector sparseness but the effective vector length is only n, the orbital basis dimension. 

R . and components Q . . The dipolar coupling between each body and each field is expressed in terms of the inner product between the field X and a unit vector u fixed on the body (so that the quantity /x u can be interpreted, if desired, as the dipole moment of the nth body) the (diagonal) matrix H has elements which measure the amplitude of the fluctuations of the components of the field X . 

The matrix product is well known and can be found in any linear algebra textbook. It reduces to the vector product when a vector is considered as an I x 1 matrix, and a transposed vector is a 1 x I matrix. The product ab is the inner product of two vectors. The product ab is called the outer product or a dyad. See Figure 2.2. These products have no special symbol. Just putting two vectors or matrices together means that the product is taken. The same also goes for products of vectors with scalars and matrices with scalars. 

For a situation with one dependent variable, a score vector is sought which has maximal covariance with y. For the two-way case the loading vector is found directly as X y/HX yH. For the three-way case the weights can also be found directly as the first left and right singular vectors from a singular value decomposition of a matrix Z (J x K) which is obtained as the inner product of X and y (see proof in Appendix 6.B). Each element in Z is given by... 

This section sets the stage for some of the ideas discussed later. Going from equations with discrete vectors and matrices to equations with functions in its basic form is not difficult. Let us demonstrate these ideas by looking at an example where PCA is applied to functions. Let X be a matrix of continuous spectra. This means that the N rows in X are really functions such that X = [xi(t) X2(t) ... xisi(t) ]. One way to find the principal components of X is to solve the eigenquation of the covariance matrix G = XX. For the discrete case G can be written in terms of vector inner products... 

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