Orthonormal eigenvectors

In all cases then, one can find n orthonormal eigenvectors (remember we required... 

We denote by Aj., the system of eigenvalues and orthonormal eigenvectors of the operator A ... 

The energy levels and MOs from these results are shown in Fig. 4.17. Note that all these matrix diagonalizations yield orthonormal eigenvectors vpVi = 1 and Vj-Vj = 0, as required the fact that the Fock matrices are symmetric (see the discussion of matrix diagonalization in Section 4.3.3). 

Let us consider the situation when the solutions (i.e., the orthonormalized eigenvectors I -0 1 and the corresponding eigenvalues E ) of the Schrodinger equation... 

The diagonalization of each Hk reduces to that of to2 — (k). Let Qks be the eigenvalues, real and positive, of yi(i(k) associated with the orthonormal eigenvectors (e ), which must satisfy... 

The columns of the matrix V are formed by the orthonormalized eigenvectors of and the diagonal elements (. ff,[Pg.121]

The matrix TT is symmetric and actually a diagonal matrix since the score vectors are orthogonal. Any real symmetric matrix can be written as the product of the transposed orthonormal eigenvector matrix, V, a diagonal matrix of eigenvalues, G, and V. 

Let Y he N X K dimensional data N < K) with mean-centered rows and columns as it is for Yr. Although the only requirement for PCA application is mean-centering of columns, having the rows mean-centered as well due to CD profile removal provides better scaling to remaining data. Define a covariance or scatter matrix Z = YY and let U = [uiU2...ujv] with Uj = be the orthonormal eigenvectors of Z such that... 

There is a complete system of common orthonormal eigenvectors ... 

We have shown that if H is a real symmetric matrix with eigenvalues Wj and orthonormal eigenvectors (that is, if for i = 1,2,..., n), then C HC =... 

Since H is symmetric, Eq. (1.164) can be solved to yield N orthonormal eigenvectors c and corresponding eigenvalues which for convenience are arranged so that q i / -1 Thus... 

The Hessian is symmetric and may therefore be diagonalized to yield a set of real eigenvalues bi associated with orthonormal eigenvectors v,. Equation (51) can therefore be represented as... 

Generally, again by an orthogonal transformation, the region (9.3.57a) is transformed to a V-dimensional ellipsoid in its standard position with respect to orthonormal eigenvectors of matrix Fj, giving... 

Liu and Klein introduced another vertex invariant defined on the basis of matrix spectral theory. Denote by X an eigenvalue of the adjacency matrix A and by jA.) the corresponding set of orthonormal eigenvectors, where x is a degeneracy index with values from 1 to the degeneracy g(X) of the eigenvalue X. Then... 

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