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Pacific Journal of Applied Mathematics - Nova Science Publishers

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A linear discrete dynamical system and its eigenvectors

The seller hasn't specified a postage method to Germany. Contact the seller - opens in a new window or tab and request post to your location. Postage cost can't be calculated. Please enter a valid postcode. There are 1 items available. Remark 1 Note that, as a consequence of the above result, the higher powers of matrix are always expressible in terms of the lower ones at most up to the dimension of.

It is well known that an analytic function f of a matrix , i. Let the spectrum of. Denoting by. If the eigenvalues are all distinct, coincides with the Lagrange interpolation polynomial and 2.


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In case of multiple eigenvalues, is the Hermite interpolation polynomial, and 2. This avoids the use of higher powers of in the Taylor expansion 2.

Pacific Journal of Applied Mathematics

In any case, the possibility to write , , in an easy block form, requires not only the knowledge of the spectrum, but even the Jordan canonical form of. It is necessary to compute the eigenvectors and moreover the principal vectors, if is defective. A known machinery which implies a lot of computations. Theorem 2 Under the hypotheses and definitions considered above, the resolvent matrix can be represented as.

Theorem 3 If is a holomorphic function in the domain and denoting by with a closed set whose boundary is a piecewise simple Jordan contour encompassing the spectrum of , the matrix function can be represented by:. Remark 2 If the eigenvalues of , are known, Equation 2. However, for computing the integrals appearing in Equation 2.

Therefore, this approach is computationally more convenient with respect to the Lagrange-Sylvester formula. As a consequence of the above recalled results, we can prove our main results both in the discrete and continuous time case. Then, taking into account the above definitions of vectors and our result follows. Remark 3 Note that, even if this is unrealistic, solution 3.

Introduce the matrix and define the vector function.


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  7. Then, Equation 3. Remark 4 Note that the convergence of the vectorial series in any compact set K of the space is guaranteed, as the components of are polynomials of weight not exceeding , and consequently are bounded in K.

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    Indeed, we can prove the following result. We show that the above results are easier with respect to the methods usually presented in literature [1] [2]. Our technique is as follows: if the matrix has a low dimension , its invariants can be computed directly by hand. If it is more easy to compute the eigenvalues by using one of the classical numerical methods, and then the invariants are found as the elementary symmetric functions with alternate sign of the eigenvalues. The Theory of Algebraic Numbers , 2 nd ed. A Modern Course on the Theory of Equations.

    Polygonal Press. Introductory Algebraic Number Theory. Let me mention several books on Galois Theory. As a rule even if some of these books do not presume a prior knowledge of group theory, you should learn some group theory before hand. The first of these books has a lot of other information and is certainly one of the best: Hadlock, Charles Robert. Field Theory and Its Classical Problems. Galois Theory , 3 rd ed. May be the best introduction. My favorite is the book by Stillwell.

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    I don't think much of it as text, but it is a great book to read. Despite the title, it is very much a book on Galois Theory. Elements of Algebra: Geometry, Numbers, Equations. Fields and Galois Theory. A Course in Galois Theory. Galois Theory. Notre Dame. Another succinct book similar to Artin's in every way is Postnikov, M. Foundations of Galois Theory.