Matrices characteristic equations

Thus, if the characteristic equation of the invariance matrix of the p order composition of (j) is... 

The methods of simple and of inverse iteration apply to arbitrary matrices, but many steps may be required to obtain sufficiently good convergence. It is, therefore, desirable to replace A, if possible, by a matrix that is similar (having the same roots) but having as many zeros as are reasonably obtainable in order that each step of the iteration require as few computations as possible. At the extreme, the characteristic polynomial itself could be obtained, but this is not necessarily advisable. The nature of the disadvantage can perhaps be made understandable from the following observation in the case of a full matrix, having no null elements, the n roots are functions of the n2 elements. They are also functions of the n coefficients of the characteristic equation, and cannot be expressed as functions of a smaller number of variables. It is to be expected, therefore, that they... 

An eigenvalue or characteristic root of a symmetric matrix A of dimension p is a root of the characteristic equation ... 

In the previous section we found that the hybrid transfer constants of a two-compartment model are eigenvalues of the transfer constant matrix K. This can be generalized to the multi-compartment model. Hence the characteristic equation can be written by means of the determinant A ... 

The second key point is the accomplishment of a desired goal on the left-hand side of equation 4-8 we have the expression [A] 1 [A], We noted earlier that the key defining characteristic of the inverse of a matrix is that fact that when multiplied by the original matrix (that it is the inverse of), the result is a unit matrix. Thus equation 4-8 is equivalent to... 

Now suppose that B is a diagonal matrix (all off-diagonal elements equal to zero) then the roots of its characteristic equation (eigenvalues) are identical with its diagonal elements. If A is not a diagonal matrix but is related to B by a similarity transformation, it follows that it has the same characteristic equation and roots as B. The problem of finding the eigenvalues... 

The second equation is called the characteristic equation of the matrix A. Once A is known, the vector ut is computed as the unit vector solution of the linear system. It is left to the reader to show that the eigenvalues of a 2 x 2 matrix A can be found as a solution of the equation... 

Solving the characteristic equation, we find that the eigenvalues of A are = —1.3289 and X2 = —37.851 with the eigenvector matrix U and its inverse given by... 

We will show in Sec. 15.3 that the eigenvalues of the 4 matrix are the roots of the characteristic equation of the system. Thus the eigenvalues tell us whether the system is stable or unstable, fast or slow, overdamped or underdamped. They are essential for the analysis of dynamic systems. 

The roots of the openloop characteristic equation are s = — 2 and s = 4, These are exactly the values we calculated for the eigenvalues of the matrix of this system ... 

Exarngde 15.15. Determine the dosedloop characteristic equation for the system whose openloop transfer function matrix was derived in Example 15.14. Use a diagonal controller structure (two SI SO.controllers) that are proportional only. 

The eigenvalues of the 4 matrix, will be the openloop eigenvalues and will be equal to the roots of the openloop characteristic equation. In order to help us... 

Thus the characteristic matrix for this closedloop system is the 4cl niatrix. Its eigenvalues will be the close oop eigenvalues, and they will be the roots of the closedloop characteristic equation. 

For openloop systems, the denominator of the transfer functions in the matrix gives the openloop characteristic equation. In Example 15.14 the denominator of the elements in was (s + 2X + 4). Therefore the openloop characteristic equation was... 

Since the matrix is not singular, its determinant is not zero. And the determinant of its inverse is also not zero. Thus the closedloop characteristic equation becomes... 

Remark f. Notice that for a linear system the coefficients ao,, ai j,..., in equation (48) represent the coefficients of the characteristic equation of matrix S. For the nonlinear case, these coefficients do not represent a generalization of the Cayley-Hamilton theorem hence the assumption is necessary for the existence of the solution of the NRRP. 

When applying the matrix method to very large values of m, it is only the largest root of the characteristic equation that is significant for calculating the thermodynamic properties of the system. For the particular matrix defined in Eq. (7.1.9), the characteristic equation is... 

The eigenvalues of A can be find by solving the characteristic equation of (1.61). It is much more efficient to look for similarity transformations that will translate A into the diagonal form with the eigenvalues in the diagonal. The Jacobi method involves a sequence of orthonormal similarity transformations, 12,... such that A(<+1 = TTkAkTk. The matrix Tk differs from the identity... 

The two roots of the matrix of the characteristic equation are 1.0688 and 19.8378. Since the smallest is larger than one, the equation is stable. 

Equation (2) is called the characteristic (or secular) equation of A, and its roots are the eigenvalues of A, ak. The problem of finding the eigenvalues of a matrix is intimately connected with its conversion to diagonal form. For if A were a diagonal matrix, then its characteristic equation would be... 

A matrix is diagonalizable if it is equivalent to a diagonal matrix D. The characteristic equation of A is invariant under a similarity transformation, for... 

In the general case, if A is not a normal matrix, then it is not necessarily diagonalizable. However, it is diagonalizable if the characteristic equation has n distinct roots. 

The denominator of eqn. (Ill) is the determinant for the weights matrix B c). But the same determinant is also a free term of the characteristic equation D that equals the product of the roots, i.e. D = n"=1A, (n is the number of independent intermediate substances). 

To interpret differences in the relaxation times, it is necessary to start from the analysis of eigenvalues of the matrix for (2)—(3) linearized in the neighbourhood of the steady state (indicated by ). This matrix corresponds to the characteristic equation... 

This equation is equivalent to the characteristic equation of the matrix (1.25) and therefore provides, when exactly resolved, all the eigenenergies of the total system in spite of the privileged role played by the matter subsystem in our calculations. We notice that an equation in z2 is effectively obtained owing to... 

The isotropic-to-nematic transition is defined by the characteristic equation Det M = 0 (where Det represents the determinant of a matrix). If the Van der Waals interactions were turned off (W0 = 0) so that only nematic interactions are left, then M would be the denominator of X so that X would blow up for this condition (Det M = 0). Above certain critical values of Wj s the blend forms the nematic phase. As in the case of purely flexible mixtures, the spinodal condition is ... 

If an eigenvalue X - as defined by the characteristic equation for T in any matrix representation - is degenerate, the situation is more complicated, and the eigenvalue problems (2.3) have to be replaced by the associated stability problems see ref. B, Sec. 4. In matrix theory, the search for the irreducible stable subspaces of T is reflected in the block-diagonalization of the matrix... 

The eigenvalues are obtained from the characteristic equation dct. / - AT] = 0, or for a 2 X 2 matrix we have the following quadratic form ... 

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