Eigenvector, definition

Thus, if we wish to compare the eigenvectors to one another, we can divide each one by equation [57] to normalize them. Malinowski named these normalized eigenvectors reduced eigenvectors, or REV". Figure 52 also contains a plot of the REV" for this isotropic data. We can see that they are all roughly equal to one another. If there had been actual information present along with the noise, the information content could not, itself, be isotropically distributed. (If the information were isotropically distributed, it would be, by definition, noise.) Thus, the information would be preferentially captured by the earliest... 

Thus far we have considered the eigenvalue decomposition of a symmetric matrix which is of full rank, i.e. which is positive definite. In the more general case of a symmetric positive semi-definite pxp matrix A we will obtain r positive eigenvalues where r general case we obtain a pxr matrix of eigenvectors V such that ... 

The above-mentioned statistical characteristics of the chemical structure of heteropolymers are easy to calculate, provided they are Markovian. Performing these calculations, one may neglect finiteness of macromolecules equating to zero elements va0 of transition matrix Q. Under such an approach vector X of a copolymer composition whose components are X = P(M,) and X2 = P(M2) coincides with stationary vector n of matrix Q. The latter is, by definition, the left eigenvector of this matrix corresponding to its largest eigenvalue A,i, which equals unity. Components of the stationary vector... 

For conservative systems with time-independent Hamiltonian the density operator may be defined as a function of one or more quantum-mechanical operators A, i.e. g= tp( A). This definition implies that for statistical equilibrium of an ensemble of conservative systems, the density operator depends only on constants of the motion. The most important case is g= [Pg.463]

As indicated in Table 4.2, the eigenvalues of the Hessian matrix of fix) indicate the shape of a function. For a positive-definite symmetric matrix, the eigenvectors (refer to Appendix A) form an orthonormal set. For example, in two dimensions, if the eigenvectors are Vj and v2, v[v2 =0 (the eigenvectors are perpendicular to each other). The eigenvectors also correspond to the directions of the principal axes of the contours of fix). 

A. 1 Definitions / A.2 Basic Matrix Operations / A.3 Linear Independence and Row Operations / A.4 Solution of Linear Equations / A. 5 Eigenvalues, Eigenvectors / References /... 

The covariance matrix is positive semi-definite and symmetric. Thus, it can be written in terms of eigenvalues and eigenvectors as... 

In space 91", S=constant is the equation of a hyper-quadric whose principal axes are colinear with the eigenvectors ut and have a half-length of A,-I/2. The simplest case occurs when all A are positive, which happens in particular when A is a product of real matrices such as B B or BBt. Then the hyper-quadric is a hyper-ellipsoid and, from the above equations, S is positive whatever the vector x. The matrix A is said to be positive definite. 

The covariance matrix is factored using the diagonal matrix A and the eigenvector matrix U as U UT. Since 5 is symmetric and positive-definite, the eigenvalues are positive and the eigenvectors orthogonal. The inverse S 1 of S can be expanded as UA 1UT and the transformation... 

As the matrix ATA is positive definite, i.e., it has positive eigenvalues, for c2 to be minimum, the solution x must be the eigenvector associated with the smallest eigenvalue Al of this matrix. 

To complete the definition of the renormalization step for the left block, we also need to construct the new matrix representations of the second-quantized operators. In the product basis Z <8> p, matrix representations can be formed by the product of operator matrices associated with left, p j and the partition orbital p separately. Then, given such a product representation of O say, the renormalized representation O in the reduced M-dimensional basis / of LEFIi. p is obtained by projecting with the density matrix eigenvectors L defined above,... 

Because diagjni,..., ni r is positive definite, the quasi-symmetric matrix L has N real eigenvalues, whereas the eigenvectors corresponding to two different eigenvalues are diag ni,..., njv -orthogonal ... 

An important application of eigenanalysis is the diagonalization of a (symmetric) matrix A. Let U denote the matrix whose columns are the normalized eigenvectors > 2<- - By the definition (1.60) we have... 

Definition 8.8 Suppose su(2 ),V, p) is a finite-dimensional Lie algebra representation. Suppose vq is an eigenvector of plfi) with the property that XpVo = 0. Then vq is a highest weight vector for the representation p. 

Next we check the eigenvector condition, Equation 8.9. By the definition of a highest weight, vq is an eigenvector for /)(i). Let Ao denote the eigenvalue of p(i) for the eigenvector vq. Then (by an easy induction) it follows from Proposition 8.7 that the eigenvalue associated to Y vo is Ao — ik. On the other hand, note that the trace of /o(i) on any finite-dimensional space is... 

Thus we take (13.1) and (13. 2) as the definition of a pseudo-eigenvalue X and a pseudo-eigenvector cp. Of course the meaning of or s must be specified in each individual case. 

The other theorem states that the matrix X formed by using the eigenvectors of a Hermitian matrix as its columns is unitary (for the definition of a unitary matrix, see Appendix A.4-l(ff)). The proof of these two theorems is given in Appendix A.4-3. 

Exercise. Prove the following lemma If H is a positive semi-definite Hermitian matrix, and F anti-Hermitian then the eigenvalues of A = H + F have nonnegative real parts. Moreover, if the real part is zero the corresponding eigenvector is an eigenvector of H and F separately. Use this lemma to show that (5.12) is the solution of (5.10). 

Note that since the matrix T is positive (semi)definite, the first K eigenvalues Tk, k - 1,. .., K, are all nonnegative. The other N - K eigenvectors of T all have zero as an eigenvalue. (Note that the nonvanishing eigenvalues of T must equal those of the K x K Hermitian matrix ft s V V.)... 

The occupation number vectors are thus the common eigenvectors for the hermitian and commuting set of operators (aj a agag,. -.a am) and there is a one to one correspondence between an occupation vector and a set of eigenvalues for (aj av a ag,- -.a am). This is consistent with the definition of the occupation number vectors as being an orthonormal basis for the Fock space. 

We now return to the question of how a left eigenvector to a matrix relates to the corresponding right eigenvector. According to the definition, a left eigenvector to a matrix B is a row vector, L, that fulfills... 

In writing this book we aimed to close this gap by taking the reader all the way from general definitions through to the detailed treatment required in specific experimental situations. In the course of this aim we arrived at a kind of three-dimensional space for the book, with pictorial illustration, strict theory and experimental examples as its eigenvectors . 

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