Matrices characteristic values

As shown in Table 10, the surface modifications cause noticeable increases of the characteristic values of composites, depending on the fiber, matrix, and type of surface treatment used. 

When used as substitutes for asbestos fibers, plant fibers and manmade cellulose fibers show comparable characteristic values in a cement matrix, but at lower costs. As with plastic composites, these values are essentially dependent on the properties of the fiber and the adhesion between fiber and matrix. Distinctly higher values for strength and. stiffness of the composites can be achieved by a chemical modification of the fiber surface (acrylic and polystyrene treatment [74]), usually produced by the Hatschek-process 75-77J. Tests by Coutts et al. [76] and Coutts [77,78] on wood fiber cement (soft-, and hardwood fibers) show that already at a fiber content of 8-10 wt%, a maximum of strengthening is achieved (Fig. 22). 

This relation is equivalent to an algebraic equation of degree n in the unknown X and therefore has n roots, some of which may be repeated (degenerate). These roots are the characteristic values or eigenvalues of the matrix B, When the determinant of Eq. (69) is expanded, the result is the polynomial equation... 

Here p is the set of characteristic values of the parameters i.e. p(x) = p co(jt) where w(x) has values centered on 1. Often we can set p = / p(x) g(x) dx. The proof is really a statement of what linearity means, for if g(x)djt is the input concentration, g(x)dx.A(p(x)) is the output when the parameter values are p(x). Here x serves merely as an identifying mark, being truly an index variable and the integration in equation (14) follows from the superposition principle for linear systems. The same principle allows an obvious extension to multiple input, multiple output linear systems. A becomes a matrix whose elements are the response at one of the out-ports to a unit input at one of the in-ports, the input at all the others being zero. The detail of this case will not be elaborated here, but rather shall we pass to an application. 

It is the eigenvalues (literally characteristic values ) of [0] that characterize the correction factor matrix [S]. Thus, the scalar rate factor 0 and correction factor S when multiplied by identity matrices frequently are quite good models for the behavior of the complete matrices [0] (or [ ]) and [H] in the exact and linearized methods. 

This differs by a factor Z from the estimate derived in Eq. VI. 7 by the assumption that the characteristic value of the transition probability matrix A can be determined by perturbation theory while the corresponding characteristic vector remains unchanged by the existence of an absorbing barrier. The significance of the error will be discussed presently. 

Givens J.W. (1954) Numerical Computation of the Characteristic Values of a Real Symmetric Matrix, Oak Ridge National Laboratory, Report ORNL-1574. 

The Gibbs energy of an atom in the ideal kink site position is the characteristic value for the molecular interaction of the atom with the bulk matrix. It takes into account bond strengths among first, second, and further neighbors. It is exactly one half of the lattice energy. 

Characteristic Values of Modified Matrix. This section will now be concluded with a theorem of considerable importance in connection with certain numerical methods for the determination of the characteristic values to be described in subsequent sections. Suppose a constant number, n, is subtracted from each of the diagonal elements of H. Then the resultant matrix may be written... 

Suppose the square of the matrix H is computed numerically. From the arguments given in Sec. 9-1, it should be evident that the characteristic values of are the squares of the characteristic values of H. Therefore... 

Procedure for Finding X2, X3, etc. The first of the two methods for this purpose which will be described here is a device which yields a new matrix which, in place of the characteristic values Xi, X2, X3,. . . , X , has the characteristic values 0, X2, X3,. . . , X . Thus, in the new matrix, X2 becomes dominant, so that iteration as above will converge upon A2 and X2. 

Another special class of matrix, the unitary matrix, A = A+, has characteristic values of absolute value unity, X X = 1. Consider its secular equation, written in matrix form... 

The characteristic vectors can be regarded as the columns of a square matrix, A. It will now be shown that the matrix A has the power to transform H into a diagonal matrix A whose diagonal values are the characteristic values X. Multiply (5) on the left by (A ) , which is the fth row (not column) of the matrix reciprocal to A. Since, by the definition of reciprocal matrices. 

The solution of the secular equation by several of the methods to be described below, Secs. 9-5 to 9-7, is appreciably simplified if H is symmetric. Moreover, in the case of one of the electric analogue devices. Sec. 9-10, which can be used to eliminate the numerical work, it is necessary that H be symmetric. Therefore a method which transforms H = GF into a symmetrical matrix with the same characteristic values as H will now be described. 

It will now be proved that the process does indeed converge upon c, = Xi, the greatest characteristic value (and upon the corresponding characteristic vector, Ai). The (unknown) characteristic vectors A are linearly independent (since they constitute the columns of a square matrix. A, which has an inverse) so that a vector with arbitrary components, such as can always be expressed as some linear combination of the Afc ... 

An alternate, and somewhat more versatile, procedure makes use of the theorem proved at the end of Sec. 9-1, namely, that a modified matrix, say XiE — H (where Xi has just been determined by iteration with H), has characteristic values 0, Xi — X2, Xj — X3,. . . , Xi — X . Iteration with XiE — H should therefore converge upon the largest of these, which is Xi — X , the characteristic vector being identical with that associated with X in H. 

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