Three combinatorial problems in matrix theory are considered in this thesis. In the first problem the structure of 0-1 matrices with permanent 1, 2, or 3, is analysed. It is shown that a 0-1 matrix whose permanent is a prime number p can be brought by permutations of rows and columns into a subdirect sum of 1 -square matrices (1) and a fully indecomposable 0-1 matrix with permanent p. The structure of fully indecomposable 0-1 matrices with permanent 1, 2, or 3, is then determined. It is found that the only fully indecomposable 0-1 matrix with permanent 1 is the 1-square matrix (1); and fully indecomposable 0-1 matrices with permanent 2 are I + K to within permutations of the rows and columns, where I is the identity matrix and K is the full cycle permutation matrix. The structure of fully indecomposable 0-1 matrices with permanent 3 is also described. In the second problem relationships are considered between two 0-1 matrices given certain connections between their corresponding principal submatrices. The matrices considered are 0-1 n-square symmetric matrices with zeros on the main diagonal. One of the theorems proved states that if two of these matrices have the same number of ones in corresponding (n - 2)-square principal submatrices, then the two matrices are identical. In the third problem the set of matrices G is determined. By definition G is the set of all n-square matrices A such that for any n-square permutation matrix P there exists a doubly stochastic matrix B such that PA = AB. It is shown that for non-singular matrices in G, the doubly stochastic matrix B must be a permutation matrix. The set of non-singular matrices in G is then determined by considering left multiplication of A by permutation matrices Pij. It is proved that G consists of the set of matrices with identical rows together with the set of matrices of the form αP + βJ where J is the n-square matrix with all entries 1, and α, β are scalars. / Science, Faculty of / Mathematics, Department of / Graduate

Let A and B be n-square complex matrices with eigenvalues λ₁, λ₂,… λn and μ₁, μ₂,…μn respectively. The matrices A and B are said to have property L if any linear combination aA + bB, with a, b complex, has as eigenvalues the numbers aλᵢ + bμᵢ, i = 1,2, …,n. A theorem of Dr. M. D. Marcus, which gives a necessary and sufficient condition such that two matrices A and B have property L in terms of the traces of various power-products of A and B, is proved. This theorem is used to investigate the conditions on B for the special cases n = 2, 3, and 4, when A is in Jordan canonical form. The final result is a theorem which gives a necessary condition on B for A and B to have property L when A is in Jordan canonical form. / Science, Faculty of / Mathematics, Department of / Graduate

If A is an n-square matrix, the p-th compound of A is a matrix whose entries are the p-th order minors of A arranged in a doubly lexicographic order . In this thesis some of the theory of compound matrices is given, including a short proof of the Sylvester-Franke theorem. This theory is used to obtain an extremum property of elementary symmetric functions of the k largest (or smallest) eigenvalues of non-negative Hermitian (n.n.h) matrices. Extensions of theorems due to Weyl and Wielandt are given. The first of these relates elementary symmetric functions of singular values of any matrix A with the same elementary symmetric functions of the eigenvalues. The second gives an extremum property of arbitrary eigenvalues of n.n.h matrices and enables inequalities relating the eigenvalues of A, B with the eigenvalues of A + B to be given (A, B, n.n.h.). Finally, a norm inequality for an arbitrary matrix is given. / Science, Faculty of / Mathematics, Department of / Graduate

Much is known about the eigenvalues of some special types of matrices. For example, the eigenvalues of a hermitian or skew-hermitian matrix lie on a line while those of a unitary matrix lie on a circle; their spectra are "linear" or "circular". This suggests the question: What matrices have this property? Or, more generally, what matrices have their eigenvalues on plane curves of a simple kind? Is it possible to recognize such matrices by inspection? In this thesis, we make a small start on these problems, exploring some matrices whose eigenvalues lie on one or more lines, or on one or more circles. / Science, Faculty of / Mathematics, Department of / Graduate

Supervisor: Dr. B. N. MOYLS Let Mm,n (F) denote the set of all mxn matrices over the algebraically closed field F of characteristic 0, and let Mn (F) denote Mn,n (F) . Let E₃ (A) denote the third elementary symmetric function of the eigenvalues of A; let Rk = {A € Mm,n (F) : rank of A = k} ; and let mA denote the minimum polynomial of A. In this paper we are concerned with those linear transformations on Mm,n (f) for which T(Rk )⊆ Rk for various k ≤ min (m,n) ; those on Mn(f) which leave E₃ invariant; and those of Mn (F) which leave the minimum polynomial invariant. The main results are as follows: If T : Mn(F) → Mn(F) and E₃(A) = E₃(T(A)) for all A ε Mn (F) where F is the field of complex numbers, then there exist nonsingular nxn matrices U and V such that either: i) T : A → UAV for all A ε Mn (F) ; or ii) T : A → UAtV for all A ε Mn (F) ; where iii) UV = eiθIn , 3θ ≡ 0 (mod 2π) . If T : Mn(F) →Mn(F) and mA = mT(A) for all A ε Mn (F) , then T has the form i) or ii) above where UV = In. Let T : Mm,n (F) → Mm,n (F) and T(Rk) ⊆ Rk. If [formulas omitted] Then there exist nonsingular mxm and nxn matrices U [formulas omitted]. / Science, Faculty of / Mathematics, Department of / Graduate

In this thesis we present the generalization of the Moore-Penrose pseudo-inverse in the sense that it satisfies the following conditions. Let x be an m × n matrix of rank r , and let u and v be symmetric positive semi-definite matrices of order m and n and rank s and t respectively, such that s.t ≥ r , and column space of x ⊂ column space of u row space of x⊂ row space of v. Then x≠ is called the generalized inverse of x with respect to u and v if and only if it satisfies : (i) xx≠x = x (ii) x≠xx≠= x≠ (iii) (xx≠)’ = u⁺xx≠u (iv) (x≠x)' = v⁺x≠xv , where U⁺ and V⁺ are the Moore-Penrose pseudo-inverses of U and V respectively. We further use this result to generalize the fundamental Gauss-Markoff theorem for linear estimation, and we also use it in the minimum mean square error estimation of the general model y = Xβ + ε , that is, we allow the covariance matrix of y to be symmetric positive semi-definite. / Science, Faculty of / Mathematics, Department of / Graduate

Let Ωn denote the convex polyhedron of all nxn d.s. (doubly stochastic) matrices. The main purpose of this thesis is to study some combinatorial properties of the diagonal sums of matrices in Ωn. In Chapter I, we determine, for all d.s. matrices unequal to Jn ; the maximum number of diagonals that can have a common diagonal sum. The key will be a Decomposition Theorem that enables us to characterize completely the structure of a d.s. matrix when this maximum number is attained, provided that the common sum is not one. When the common sum is one, the question is more difficult and remains open. Several applications of the Decomposition Theorem are also given. In Chapter II, we concentrate on the diagonals with maximum diagonal sum h and the diagonals with minimum diagonal sum k. We obtain the best possible bounds for entries on these diagonals and for various kinds of functions of h and k. The key will be a Covering Theorem that enables us to analyze the cases when those bounds are attained. A conjecture is given. In Chapter III, we study the properties of the h-function and the k-function, the functions that associate with each d.s. matrix its maximum and minimum diagonal sums respectively. In particular, we investigate the behavior of these functions on the Kronecker product of d.s. matrices. Furthermore, we show that the h-function is very similar to the rank function ƿ in many respects. We also prove that for A ε Ωn, h(A) ≤ ƿ(A) and per (A) ≤ [formula omitted] which improves a result by M. Marcus and H. Mine. A conjecture is given. / Science, Faculty of / Mathematics, Department of / Graduate

Let Sn denote the set of n x n permutation matrices; let T denote the set of transpositions in Sn; let C denote the set of 3-cycles {(r, r+1, t) ; r = 1, …, n-2; t = r+2, …, n} and let I denote the identity matrix in Sn. We shall denote the n-lst elementary symmetric function of the eigenvalues of A by [formula omitted]. In this thesis, we pose the following problems: 1. Let H be a subset of Sn and a1, …, ak be k-distinct real numbers. Determine the set of n-square matrices A such that {tr(PA):P є H} = {a1, ..., ak} . We examine the cases when (i) H = Sn, k = 1 / (ii) H = {2-cycles in Sn} , k = 1 (iii) H = Sn, k = 2 2. Determine the set of n x. n matrices such that [formula omitted]. 3. Examine those orthogonal matrices which can be expressed as linear combinations of permutation matrices. The main results are as follows: If R’ is the subspace of rank 1 matrices with all rows equal and if C’ is the subspace of rank 1 matrices with all columns equal, then the n x n matrices A such that tr(PA) = tr(A) for all P є Sn form a subspace S = R' + C’.This implies- that the.' rank of A is ≤ 2. If tr(PA) = tr(A) for all P є T , then such A's form a subspace which contains all n x n skew-symmetric matrices and is of dimension [formula omitted]. Let A be an n-square matrix such that (tr(PA) : P є Sn} = {a1, a.2} , where a1 ≠ a2. Then A is either of the form C = A1 + A2, where A1 є (R’ +. C’) and A2 has entries a1 – a2 at [formula omitted], j =2, …, k and zeros elsewhere, or of the form CT. The set [formula omitted] consists of 1 n 1 all 2-cycles (r^, rj)> j = 2, ..., k and the products P of disjoint cycles [formula omitted], for which one of the P1 has its graph with an edge [formula omitted]. If A is rank n-1 n-square matrix with the property That[formula omitted] for all P є Sn, then A is of the form [formula omitted] where Ui are the row vectors. Finally,.if [formula omitted] , where all P1, are from an independent set TUCUI of Sn, is an orthogonal matrix, then [formula omitted]. / Science, Faculty of / Mathematics, Department of / Graduate

The purpose of this paper is the study of congruent matrices over fields and valuation rings. / Advisor: Paul J. McCarthy, Professor Directing Paper. / Typescript. / "January, 1959." / "Submitted to the Graduate Council of Florida State University in partial fulfillment of the requirements for the degree of Master of Science." / Includes bibliographical references (leaf 16).

Thesis (Ph. D.)--Washington State University, December 2009. / Title from PDF title page (viewed on Dec. 28, 2009). "Department of Mathematics." Includes bibliographical references (p. 101-104).