Dear all,
I would be very grateful if someone could direct me towards some literature that studies matrices with polynomial entries, especially matrix norms. I am dealing with matrices whose elements are polynomials. These can also be thought of as polynomials whose coefficients are matrices.
Kind regards,
Salem.
Unfortunatelly, I have worked very little with it. Perhaps the closest I've gotten is that paper I published in Comm. Alg. in 2002. Maybe you can be interested in: higher derivations in semiprime rings, available at my RG spot. A hug, C.
I suggest that a good habit is first to search internet. I just did it for you.
Polynomial matrix
http://en.wikipedia.org/wiki/Polynomial_matrix
E. Horowitz On Computing the Exact Determinant of Matrices with Polynomial Entries, Journal of the ACM (JACM) Volume 22 Issue 1, Jan. 1975
www.nyu.edu/classes/edwards/chap8.html
I do not see any reason why should the approach be different between matrices of constant coefficients and polynomial coefficients. For example: the determinant is a constant in the first case and a polynomial in the second case.
I would like to suggest a good book for you. In Chapter 6 of the following book.
Charles G. Cullen, Matrices and Linear Transformations, Dover Publications, Inc., New York, 1972.
Dear Abedallah M Rababah,
I don't think the point you raised is correct for all aspects of the matrix. Sure what you said about the determinant is true. Other things, however, are different. For instance, when you look at the norm of a matrix with polynomial entries, it is not exactly the same as that of a scalar matrix. The difference is almost like the difference between the norm of a vector and a scalar matrix. To further elaborate the point, take the frobenius norm. For a polynomial matrix, it a third summation is incurred. However, I don't really understand what the conditions of this polynomial matrix norm are.
Gohberg, Lancaster, Rodman, Dopico, de Teran, Tisseur, Meerbergen are just a few of the many authors who have written about this subject. There are many more.
Do a simple internet search and you will get more information.
I would like to suggest the quite recent paper available as preprint at arXiv https://arxiv.org/pdf/1511.03730.pdf
(63 pages, January 23, 2019) which contains Appendix “A Symbolic matrices with polynomial entries and non-commutative rank”
As Prof. Aaron Melman has mentioned, you can read the book of Gohberg, Lancaster, Rodman for matrix polynomials in one variable. For multivariate matrix polynomials, there are some works of J. Cimpric, Zalar, M. Schweighofer, K. Schmuedgen, Y. Savchuk, I. Klep,... See also from my project: https://www.researchgate.net/project/Some-selected-topics-on-matrix-polynomials.
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