Properties of Eigenvectors

If L is diagonal, TLT-1 is also diagonal, containing the same values. We can think of this process as expressing the matrix with respect to a different coordinate system. 

What happens when an eigenvector is multiplied by a matrix  

We consider first the case where L is a diagonal matrix, so that there are no off-diagonal entries. The more general case will be considered on page 22 below. 

Thus the effect of multiplication by the matrix on an eigenvector is to multiply that vector by the corresponding eigenvalue. 

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So now we understand that when we use eigenvectors to define an "abstract factor space that spans the data," we aren t changing the data at all, we are simply finding a more convenient coordinate system. We can then exploit the properties of eigenvectors both to remove noise from our data without significantly distorting it, and to compress the dimensionality of our data without compromising the information content. 

Bonacich, P. (2007) Some unique properties of eigenvector centrality. Social Networks, 29, 555-564. 

Eigenvectors and PCA. This is the factor space we are about to explore. We will be working with a factor space defined by the eigenvectors of our data simply because a coordinate system of eigenvectors has certain properties that are convenient and valuable to us. 

D.—There exists a set of three operators, Qk, h — 1,2,3 corresponding to the measurement of position q = (x,y,z). There exists a continuum of eigenvectors of these operators, q>, with the following normalization properties (cf. Eq. (8-32)) ... 

Let us dwell on the properties of eigenvalues and eigenvectors of a linear self-adjoint operator A. A number A such that there exists a vector 0 with = A is called an eigenvalue of the operator A. This vector... 

Let us now consider a given contribution to ,w) proportional to Xr and involving d clusters. We associate with it a diagram with r — 1 intermediate states. On account of the formal property of the eigenvectors... 

Multiscale ensembles of reaction networks with well-separated constants are introduced and typical properties of such systems are studied. For any given ordering of reaction rate constants the explicit approximation of steady state, relaxation spectrum and related eigenvectors ( modes ) is presented. In particular, we prove that for systems with well-separated constants eigenvalues are real (damped oscillations are improbable). For systems with modular structure, we propose the selection of such modules that it is possible to solve the kinetic equation for every module in the explicit form. All such solvable networks are described. The obtained multiscale approximations, that we call dominant systems are... 

We can now list some of the most important properties of the various types of operators and matrices Any hermitean, antihermitean, unitary, or idempotent operator has a spectral resolution where the eigenvectors form an ON-basis, so that... 

Thus the dispersion relation is symmetric about q = 0. It follows from eqs. (18) and (21) that the components of c ( q/) satisfy the same set of 3.v linear homogeneous equations as the components of the eigenvector c(q /). Therefore, if degeneracy is absent, e(q /) and c ( q /) can only differ by a phase factor (which preserves normalization). The physical properties of the system are independent of the choice of this phase factor, which we take to be... 

The evolution of this determinant first yields the eigenvalues. The solution of the whole eigenvalue problem provides pairs of eigenvalues and eigenvectors. The mathematical algorithm is described in detail in [MALINOWSKI, 1991]. A simple example, discussed in Section 5.4.2, will demonstrate the calculation. The following properties of these abstract mathematical measures are essential ... 

Some Properties of Coherent States Expansion of the Coherent State on the Eigenvectors of the Quantum Harmonic Oscillator Hamiltonian... 

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