Eigenvectors equations

We also know that if there exists an equilibrium distribution, p, then it must remain invariant under the Markovian transition i.e. p o must satisfy the eigenvector equation... 

In this example, we are analysing the data set with the low noise level and, accordingly, the distinction between structured and noise residuals is crisp and unambiguous. Only noise is left after the subtraction of the contributions of three eigenvectors, equations (5.9) and (5.11). 

The set of eigenvalue-eigenvector equations has non-trivial (v(k) = 0 is "trivial") solutions... 

We want an eigenvalue equation because (cf. Section 4.3.4) we hope to be able to use the matrix form of a series of such equations to invoke matrix diagonalization to get eigenvalues and eigenvectors. Equation (5.35) is not quite an eigenvalue equation, because it is not of the form operation on function = k x function, but rather operation on function = sum of (k x functions). However, by transforming the molecular orbitals to a new set the equation can be put in eigenvalue form (with a caveat, as we shall see). Equation 5.35 represents a system of equations... 

This has an exponential of a matrix. It is defined in terms of the expansion of the exponential function, see for example Smith [514, p. 134-5]. Now, the usual eigenvalue-eigenvector equation can be written in compact form,... 

PROPERTIES OF EIGENVALUES AND EIGENVECTORS Equation (1.44) is easily verified ... 

Note that this procedure can be used only if all the eigenvalues are real and distinct. Also, for obtaining the eigenvectors (equation 5.26) since j8s are coupled, all of the equations are solved simultaneously. In chapter 5.1, the equations for y s were solved individually one by one. 

It is sometimes convenient to consider a matrix as consisting of sub-blocks, each of which is another matrix of smaller dimension. Consider the eigenvector equation, Eq. (9), for a Hermitian matrix that is partitioned into four sub-blocks... 

The vector i 7) is a state vector whose components are all zero except for the last, or Nth component, which is taken as 1. It is evident that equation (4.96) is identical to the eigenvector equation (4.94) except at one of the boundaries, and that f ) can therefore be thought of as a particular solution. The vector may be expressed formally as... 

The original Y-matrix is approximated by an equivalent model. The third relationship describes a model for correlating scores from Y and scores from X. yB is a diagonal matrix containing the regression coefficient of the PLS component pairs. The required PLS components can be calculated as eigenvectors (equations 31 and 32) ... 

---