Column-orthonormal

The 3x3 matrix U shown below is both row- and column-orthonormal ... 

We then allow Ri and R2 to vary, subject to orthonormality, just as in the closed-shell case. Just as in the closed-shell case, Roothaan (1960) showed how to write a Hamiltonian matrix whose eigenvectors give the columns U] and U2 above. 

This procedure would generate the density amplitudes for each n, and the density operator would follow as a sum over all the states initially populated. This does not however assure that the terms in the density operator will be orthonormal, which can complicate the calculation of expectation values. Orthonormality can be imposed during calculations by working with a basis set of N states collected in the Nxl row matrix (f) which includes states evolved from the initially populated states and other states chosen to describe the amplitudes over time, all forming an orthonormal set. Then in a matrix notation, (f) = (f)T (t), where the coefficients T form IxN column matrices, with ones or zeros as their elements at the initial time. They are chosen so that the square NxN matrix T(f) = [T (f)] is unitary, to satisfy orthonormality over time. Replacing the trial functions in the TDVP one obtains coupled differential equations in time for the coefficient matrices. 

The columns of the orthonormal matrix Vp are linear combinations of reaction invariants . In fact, the only invariants for the batch reaction being analyzed can be stoichiometric coefficients. Hence the matrix Vp may be interpreted as containing the stoichiometric information (Waller and Makila (1981)) and its rank Nr can be considered to be equal to the number of independent... 

Equation (31.3) defines the eigenvalue decomposition (EVD), also referred to as spectral decomposition, of a square symmetric matrix. The orthonormal matrices U and V are the same as those defined above with SVD, apart from the algebraic sign of the columns. As pointed out already in Section 17.6.1, the diagonal matrix can be derived from A simply by squaring the elements on the main diagonal of A. 

We have seen above that the r columns of U represent r orthonormal vectors in row-space 5". Hence, the r columns of U can be regarded as a basis of an r-dimensional subspace 5 of 5". Similarly, the r columns of V can be regarded as a basis of an r-dimensional subspace S of column-space 5. We will refer to S as the factor space which is embedded in the dual spaces S" and SP. Note that r

factor-spaces will be more fully developed in the next section. 

In the previous section we have developed principal components analysis (PCA) from the fundamental theorem of singular value decomposition (SVD). In particular we have shown by means of eq. (31.1) how an nxp rectangular data matrix X can be decomposed into an nxr orthonormal matrix of row-latent vectors U, a pxr orthonormal matrix of column-latent vectors V and an rxr diagonal matrix of latent values A. Now we focus on the geometrical interpretation of this algebraic decomposition. 

Once we have obtained the projections S and L of X upon the latent vectors V and U, we can do away with the original data spaces S and 5". Since V and U are orthonormal vectors that span the space of latent vectors each row i and each column j of X is now represented as a point in as shown in Figs. 31.2c and d. The... 

In this way, an nxpxq table X is decomposed into an rxsxt core matrix Z and the nxr, pxs, qxt loading matrices A, B, C for the row-, column- and layer-items of X. The loading matrices are column-wise orthonormal, which means that ... 

To summarize, suppose that all possible states for any given observable (spin, polarization, energy, momentum, etc.) are known and that each can be formulated in terms of a column vector a = a,i, <22, an. These vectors form an orthonormal set and are represented by an n x n matrix... 

If columns (or rows) of X are normalised to the square root of the sum of their squared elements (i.e. to unity length), the matrix is called orthonormal. Recall that earlier this kind of normalisation was solved most elegantly by right (left) multiplication with a diagonal matrix comprising the appropriate normalisation coefficients. See the section introducing diagonal matrices for more details. 

If matrix X is square and has orthonormal rows, its columns are also orthonormal. The inverse is then equal to its transpose... 

Now the expression Normal Equations starts to make sense. The residual vector r is normal to the grey plane and thus normal to both vectors f ,i and f , 2 As outlined earlier, in Chapter Orthogonal and Orthonormal Matrices (p.25), for orthogonal (normal) vectors the scalar product is zero. Thus, the scalar product between each column of F and vector r is zero. The system of equations corresponding to this statement is ... 

SVD is completely automatic. It is one of the most stable algorithms available and thus can be used blindly. It is one command in Matlab [U, S, Vt]=svd (Y, 0). The matrices U and V1 contain as columns so-called eigenvectors. They are orthonormal (see Orthogonal and Orthonormal Matrices, p.25) which means that the products... 

The elements of D represent the sum over all unit cells of the interaction between a pair of atoms. D has 3n x 3n elements for a specific q and j, though the numerical value of the elements will rapidly decrease as pairs of atoms at greater distances are considered. Its eigenvectors, labeled e ( fcq), where k is the branch index, represent the directions and relative size of the displacements of the atoms for each of the normal modes of the crystal. Eigenvector ejj Icq) is a column matrix with three rows for each of the n atoms in the unit cell. Because the dynamical matrix is Hermitian, the eigenvectors obey the orthonormality condition... 

As we have pointed out many times previously, the columns of the standard tableaux functions are antisymmetrized, and the orbitals in a column may be replaced by any linear combination of them with no more than a change of an unimportant overall constant. In this case, consider a linear combination that has two hybrid orbitals that point directly at the H atoms in accord with Pauling s principle of maximum overlap. Using the parameter

[Pg.180]

Recall from linear algebra that orthogonal matrices have columns that form an orthonormal basis. Orthogonal linear operators preserve the Euclidean structure, i.e., if we let a dot denote the EucUdean dot product we have... 

If btb= 1, then b is said to be normalized. If a set of column vectors has every member normalized and every member orthogonal to every other member, the set is called orthonormal. (Orthonormality was previously used to describe functions we shall see in the next section that orthonormal functions can be represented by orthonormal column vectors.)... 

Note from (2.62) that orthonormal functions have orthonormal column vector representatives. 

The first equation in (2.24) states that column vectors / and j of a unitary matrix are orthonormal the second equation states that the row vectors of a unitary matrix form an orthonormal set. For a real orthogonal matrix, the row vectors are orthonormal, and so are the column vectors. 

Let f and g be the column vectors representing the functions / and g in some orthonormal basis and let A represent the operator A in that basis. The reader can verify (Problem 2.22) that the integrals and [Pg.303]

Since F is a Hermitian operator, F is a Hermitian matrix and its eigenvectors can be chosen as orthonormal. Let C be the unitary square matrix of column eigenvectors of F ... 

Since the column vectors of B are orthonormal, B is a unitary matrix [Equation (2.24)]. The localized and canonical MOs are related by a unitary transformation. Since B is unitary, we can write (2.81) as can... 

The remaining columns follow from the group multiplications. It will now be shown that these representations satisfy the orthonormalization condition of 4.5-1. 

In the unlikely event that none of the basis functions overlap, then S is a unit matrix. We usually require the LCAO orbitals ipA, Pb, , V m to be orthonormal and this fact can be summarized in a single matrix statement. A little manipulation will show that UTSU is then a unit matrix (with m rows and m columns), and also that... 

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