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Nonnegative Matrices and Stochastic Matrices
Published in Leslie Hogben, Richard Brualdi, Anne Greenbaum, Roy Mathias, Handbook of Linear Algebra, 2006
An n × nmatrix polynomial of degreed in the (integer) variable m is a polynomial in m with coefficients that are n × n matrices (expressible as S(m)=∑t=0dmtBt with B1, …, Bd as n × n matrices and Bd ≠ 0).
Matrices and Linear Algebra
Published in William S. Levine, Control System Fundamentals, 2019
For any polynomial, p(s) = p0sk +p1sk−1+⋯+pk–1s+pk, with coefficients pi ∈ R, the matrix polynomial p(A) is defined as p(A) = p0Ak + p1Ak−1 + ⋯ + pk−1A + pkI. When the ring of scalars, R, is a field (and in some more general cases), n × n matrices obey certain polynomial equations of the form p(A) = 0; such a polynomial p(s) is an annihilating polynomial of A. The monic annihilating polynomial of least degree is called the minimal polynomial of A; the minimal polynomial is the (monic) greatest common divisor of all annihilating polynomials. The degree of the minimal polynomial of an n × n matrix is never larger than n because of the remarkable Cayley-Hamilton Theorem. Let A ∈ Rn×n, where R is a field. Let χ (s) be the nth degree monic polynomial defined by () χ(s)=det(sI−A)Then, χ (A) = 0.
Solution of the symmetric band partial inverse eigenvalue problem for the damped mass spring system
Published in Inverse Problems in Science and Engineering, 2021
Suman Rakshit, Biswa Nath Datta
The dynamics of such a system are governed by the eigenvalues and eigenvectors of the associated quadratic matrix polynomial . When M is non-singular which is true for most practical problems, the polynomial has eigenvalues and eigenvectors each of length n, assuming that all these are of order n. Theoretically, the quadratic eigenvalue problem for can be solved by transforming the problem to a standard or a generalized linear eigenvalue problem. The state of the art computational methods are capable of computing only a few extremal eigenvalues and eigenvectors. For details, see [1,2].