Linear operations

This should apply both to linear and antilinear operators, hereafter denoted, respectively, by t and A. For a linear operator, the action on a general state ci /i) + C2 v /2) is expressed by... 

Plijcc the orij ini)l eoeffieient matrix next to the unit matrix. Divide row 1 by 2 ami subtraet the result from row 2, This is a linear operation, and it does not ehange the solution set. 

It is easy to see that we ean form the matrix representation of any linear operator for any eomplete basis in any spaee. To do so, we aet on eaeh basis funetion with the operator and express the resulting funetion as a linear eombination of the original basis funetions. The eoeffieients that arise when we express the operator aeting on the funetions in terms of the original funetions form the the matrix representation of the operator. 

The approximate method developed is constructive in the following sense. If A is a linear operator, then the equation (1.105) is linear too and, therefore, it can be solved by standard numerical methods. 

Now let us consider the second presentation of the variational inequality (1.126) by means of the projection operators. Suppose that A is a linear operator such that... 

Log arithmic-Mean Driving Force. As noted eadier, linear operating lines occur if all concentrations involved stay low. Where it is possible to assume that the equiUbrium line is linear, it can be shown that use of the logarithmic mean of the terminal driving forces is theoretically correct. When the overall gas-film coefficient is used to express the rate of absorption, the calculation reduces to solution of the equation... 

There are important figures of merit (5) that describe the performance of a photodetector. These are responsivity, noise, noise equivalent power, detectivity, and response time (2,6). However, there are several related parameters of measurement, eg, temperature of operation, bias power, spectral response, background photon flux, noise spectra, impedance, and linearity. Operational concerns include detector-element size, uniformity of response, array density, reflabiUty, cooling time, radiation tolerance, vibration and shock resistance, shelf life, availabiUty of arrays, and cost. 

Groetsch, C. W. Generalized Inverses of Linear Oper ators, Marcel Dekker, New York (1977). 

Naylor, A. W, and G. R. Sell. Linear Operator Theory in Engineering and Science, Springer-Verlag, New York (1982). 

Column type Exclusion limit (polystyrene equivalent) Linear operating range (polystyrene equivalent)... 

Application of the definition shows that the Laplace transform is a linear oper-ator " this property is represented in Eqs. (3-67) and (3-68). 

Kato, T, 1980, Perturbation Theory for Linear Operators , Springer-Verlag, Berlin. 

Thus, the remainder operator, R, is a linear operator that annihilates every function t of the basic set to be used in the approximation. 

The form of Lz in Cartesian coordinates is Eq. (7-la), and it is clear that orbital angular momentum is related to angular displacement in the same way as the linear operators are related. 

C. —To every measurable property of a system there corresponds a linear operator in Stf. If the measurement of the property corresponding to the operator L is performed on a system always initially prepared in the normalized state Q at time t, the mean value of the result of a series of such repeated measurements is... 

Being a product of delta functions, it is an improper expression, and the trace is undefined. However, it is useful to consider the operator product P(X)P where R is any linear operator defined in configuration space. The typical element of the matrix product is... 

Let R be any linear operator in occupation number space, and consider the product WNR. A typical matrix element of this operator is... 

Consider the product of W9 and an arbitrary linear operator B. The typical matrix element is... 

Entropy and Equilibrium Ensembles.—If one can form an algebraic function of a linear operator L by means of a series of powers of L, then the eigenvalues of the operator so formed are the same algebraic function of the eigenvalues of L. Thus let us consider the operator IP, i.e., the statistical matrix, whose eigenvalues axe w ... 

For any linear operator 22 defined in Fock space, we can similarly prove, by following an argument like that leading to Eq. (8-189), that the trace in Fock space of WB is the grand-ensemble-average of 22 ... 

From the definitions (a)-(b) it follows that a product of an even number of antilinear operators is a linear operator, whereas the product of an odd number of antilinear operators is an antilinear operator. Similarly a product of any number of linear operators and an even (odd) number of antilinear operators is a linear (antilinear) operator. 

If we assume that the adjoint A of an antilinear operator is defined as in the case of a linear operator by the equation... 

Linear algebraic problem, 53 Linear displacement operator, 392 Linear manifolds in Hilbert space, 429 Linear momentum operator, 392 Linear operators in Hilbert space, 431 Linear programming, 252,261 diet problem, 294 dual problem, 304 evaluation of methods, 302 in matrix notation, simplex method, 292... 

Our account of the theory of difference schemes is mostly based on elementary notions from functional analysis. In what follows we list briefly widespread tools adopted in the theory of linear operators which will be used in the body of this book. 

Linear operators. Let X and Y be normed vector spaces and T be a subspace of the space X. If to each vector x V there corresponds by an... 

It is worth noting here that in a finite-dimensional space any linear operator is bounded. All of the linear bounded operators from X into Y constitute what is called a normed vector space, since the norm j4 of an operator A satisfies all of the axioms of the norm ... 

Theorem 1 Let A be a linear operator from X into Y. In order that the inverse operator A exist and be bounded, as an operator from Y into X, it is necessary and sufficient that there is a constant 6 > 0 such that for all X EX... 

Linear bounded operators in a real Hilbert space. Let H he a real Hilbert space equipped with an inner product x,y) and associated norm II X II = (x, x). We consider bounded linear operators defined on the space... 

It is straightforward to verify that the relation established on the set of linear operators (H H) possesses the following properties ... 

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