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The longstanding nonnegative inverse eigenvalue problem (NIEP) is to determine which multisets of complex numbers occur as the spectrum of an entry-wise nonnegative matrix. Although there are some well-known necessary conditions, a solution to the NIEP is far from known. An invertible matrix is called a Perron similarity if it diagonalizes an irreducible, nonnegative matrix. Johnson and Paparella [Linear Algebra Appl. 493 (2016), pp. 281–300] developed the theory of real Perron similarities. Here, we fully develop the theory of complex Perron similarities. Each Perron similarity gives a nontrivial polyhedral cone and convex polytope of realizable spectra (thought of as vectors in complex Euclidean space). The extremals of these convex sets are finite in number, and their determination for each Perron similarity would solve the diagonalizable NIEP, a major portion of the entire problem. By considering Perron similarities of certain realizing matrices of Type I Karpelevič arcs, large portions of realizable spectra are generated for a given positive integer. This is demonstrated by producing a nearly complete geometrical representation of the spectra of four-by-four stochastic matrices. Similar to the Karpelevič region, it is shown that the subset of complex Euclidean space comprising the spectra of stochastic matrices is compact and star-shaped. Extremal elements of the set are defined and shown to be on the boundary. It is shown that the polyhedral cone and convex polytope of the discrete Fourier transform (DFT) matrix corresponds to the conical hull and convex hull of its rows, respectively. Similar results are established for multifold Kronecker products of DFT matrices and multifold Kronecker products of DFT matrices and Walsh matrices. These polytopes are of great significance with respect to the NIEP because they are extremal in the region comprising the spectra of stochastic matrices. Implications for further inquiry are also given.
51The nonnegative inverse eigenvalue problem from the coefficients of the characteristic polynomial. EBL digraphs
The nonnegative inverse eigenvalue problem from the coefficients of the characteristic polynomial. EBL digraphs
12Normal nonnegative realization of spectra
The nonnegative inverse eigenvalue problem is the problem of finding necessary and sufficient conditions for the existence of an entrywise nonnegative matrix A with prescribed spectrum. This problem remains open for . If the matrix A is required to be normal, the problem will be called the normal nonnegative inverse eigenvalue problem (NNIEP). Sufficient conditions for a list of complex numbers to be the spectrum of a normal nonnegative matrix were obtained by Xu [Linear Multilinear Algebra. 1993;34:353–364]. In this paper, we give a normal version of a rank-r perturbation result due to Rado and published by Perfect [Duke Math. J. 1955;22:305–311], which allow us to obtain new sufficient conditions for the NNIEP to have a solution. These new conditions significantly improve Xu’s conditions. We also apply our results to construct nonnegative matrices with arbitrarily prescribed elementary divisors.
Read moreAbout the persymmetric nonnegative inverse eigenvalue problem in low dimension
About the persymmetric nonnegative inverse eigenvalue problem in low dimension
2On the realizability of the critical points of a realizable list
On the realizability of the critical points of a realizable list
5On nonnegative matrices with prescribed eigenvalues and diagonal entries
On nonnegative matrices with prescribed eigenvalues and diagonal entries
2A Note on NIEP for Leslie and Doubly Leslie Matrices
The nonnegative inverse eigenvalue problem (NIEP) consists of finding necessary and sufficient conditions for the existence of a nonnegative matrix with a given list of complex numbers as its spectrum. If the matrix is required to be Leslie (doubly Leslie), the problem is called the Leslie (doubly Leslie) nonnegative eigenvalue inverse problem. In this paper, necessary and/or sufficient conditions for the existence and construction of Leslie and doubly Leslie matrices with a given spectrum are considered.
Read more33A map of sufficient conditions for the symmetric nonnegative inverse eigenvalue problem
A map of sufficient conditions for the symmetric nonnegative inverse eigenvalue problem
62An inequality for nonnegative matrices and the inverse eigenvalue problem
We present two versions of the same inequality, relating the maximal diagonal entry of a nonnegative matrix to its eigenvalues. We demonstrate a matrix factorization of a companion matrix, which leads to a solution of the nonnegative inverse eigenvalue problem (denoted the nniep) for 4×4 matrices of trace zero, and we give some sufficient conditions for a solution to the nniep for 5×5 matrices of trace zero. We also give a necessary condition on the eigenvalues of a 5×5 trace zero nonnegative matrix in lower Hessenberg form. Finally, we give a brief discussion of the nniep in restricted cases. Keywords: Nonnegative matricesMaximal diagonal entrynniepCompanion matricesAMS Subject Classification: 15A1815A2315A42
Read more73Exponential convergence of products of stochastic matrices
Exponential convergence of products of stochastic matrices
968Geophysical Inverse Theory and Regularization Problems
Geophysical Inverse Theory and Regularization Problems
2A note on the nonzero spectra of irreducible matrices
In this note we extend the necessary and sufficient conditions of Boyle–Handleman [M. Boyle and D. Handelman, The spectra of nonnegative matrices via symbolic dynamics, Ann. Math. 133 (1991), pp. 249–316] and Kim–Ormes–Roush [K.H. Kim, N.S. Ormes, and F.W. Roush, The spectra of nonnegative integer matrices via formal power series, J. Am. Math. Soc. 13 (2000), pp. 773–806] for a nonzero eigenvalue multiset of primitive matrices over ℝ+ and Z +, respectively, to irreducible matrices.
Read moreDecomposition of matrices into product of idempotents and separativity of regular rings
Following O'Meara's result [Journal of Algebra and Its Applications, 13 (2014) 8], it follows that the block matrix A = B 0 0 0 M n+r (R), B M n (R), r 1, over a von Neumann regular separative ring R, is a product of idempotent matrices.Furthermore, this decomposition into idempotents of A also holds when B is an invertible matrix and R is a GE ring (defined by Cohn [New mathematical monographs: 3, Cambridge University Press, 2006]).As a consequence, it follows that if there exists an example of a von Neumann regular ring R over which the matrix A = B 0 0 0 M n+r (R), where B M n (R), r 1, cannot be expressed as a product of idempotents, then R is not separative, thus providing an answer to an open question whether there exists a von Neumann regular ring which is not separative.The paper concludes with a counter example related to an open question whether every totally nonnegative matrix is a product of nonnegative idempotent matrices.
Read more10On universal realizability of spectra
On universal realizability of spectra
Orthogonal Discrete Fourier and Cosine Matrices for Signal Processing
The A DFT (Discrete Fourier Transform) has seen studied and applied to signal processing and communication theory. The relation between the Fourier matrix and the Hadamard transform was developed in [Ahmed & Rao, 1975; Whelchel & Guinn, 1968] for signal representation and classification and the Fast Fourier-Hadamand Transform(FFHT) was proposed. This idea was further investigated in [Lee & Lee, 1998] as an extension of the conventional Hadamard matrix. Lee et al [Lee & Lee, 1998] has proposed the Reverse Jacket Transform(RJT) based on the decomposition of the Hadamard matrix into the Hadamard matrix(unitary matrix) itself and a sparse matrix. Interestingly, the Reverse Jacket(RJ) matrix has a strong geometric structure that reveals a circulant expansion and contraction properties from a basic 2x2 sparse matrix. The discrete Fourier transform (DFT) is an orthogonal matrix with highly practical value for representing signals and images [Ahmed & Rao, 1975; Lee, 1992; Lee, 2000]. Recently, the Jacket matrices which generalize the weighted Hadamard matrix were introduced in [Lee, 2000], [Lee & Kim, 1984, Lee, 1989, Lee & Yi, 2001; Fan & Yang, 1998]. The Jacket matrix1 is an abbreviated name of a reverse Jacket geometric structure. It includes the conventional Hadamard matrix [Lee, 1992; Lee, 2000; Lee et al., 2001; Hou et. al., 2003], but has the weights,ω , that are j or 2 , where k is an integer, and 1 j = − , located in the central part of Hadamard matrix. The weighted elements' positions of the forward matrix can be replaced by the non-weighted elements of its inverse matrix and the signs of them do not change between the forward and inverse matrices, and they are only as element inverse and transpose. This reveals an interesting complementary matrix relation. Definition 1: If a matrix m J ⎡ ⎤ ⎣ ⎦ of size m m × has nonzero elements
Read more2Factorizations of $k$-nonnegative matrices
A matrix is $k$-nonnegative if all its minors of size $k$ or less are nonnegative. We give a parametrized set of generators and relations for the semigroup of $k$-nonnegative $n\times n$ invertible matrices in two special cases: when $k = n-1$ and when $k = n-2$, restricted to unitriangular matrices. For these two cases, we prove that the set of $k$-nonnegative matrices can be partitioned into cells based on their factorizations into generators, generalizing the notion of Bruhat cells from totally nonnegative matrices. Like Bruhat cells, these cells are homeomorphic to open balls and have a topological structure that neatly relates closure of cells to subwords of factorizations. In the case of $(n-2)$-nonnegative unitriangular matrices, we show the cells form a Bruhat-like CW-complex.
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