Spelling suggestions: "subject:"amatrices"" "subject:"2matrices""
1 
Generalized matrix inverses and the generalized GaussMarkoff theoremAng , SiowLeong January 1971 (has links)
In this thesis we present the generalization of the MoorePenrose pseudoinverse 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 semidefinite 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 MoorePenrose pseudoinverses of U and V respectively. We further use this result to generalize the fundamental GaussMarkoff 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 semidefinite. / Science, Faculty of / Mathematics, Department of / Graduate

2 
Some combinatorial properties of the diagonal sums of doubly stochastic matricesWang, Edward TzuHsia January 1971 (has links)
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 hfunction and the kfunction, 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 hfunction 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

3 
Matrices which, under row permutations, give specified values of certain matrix functionsKapoor, Jagmohan January 1970 (has links)
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 3cycles {(r, r+1, t) ; r = 1, …, n2; t = r+2, …, n} and let
I denote the identity matrix in Sn. We shall denote the nlst
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 kdistinct real numbers. Determine the set of nsquare matrices A such that {tr(PA):P є H} = {a1, ..., ak} . We examine the cases
when
(i) H = Sn, k = 1 /
(ii) H = {2cycles 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 skewsymmetric matrices and is of dimension [formula omitted].
Let A be an nsquare 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 2cycles (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 n1 nsquare 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

4 
Matrices with linear and circular spectraChang, LuangHung January 1969 (has links)
Much is known about the eigenvalues of some special types of matrices. For example, the eigenvalues of a hermitian or skewhermitian 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

5 
Linear transformations on matrices : the invariance of certain matrix functionsBeasley, Leroy B January 1969 (has links)
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

6 
Classes of unimodular integral symmetric positive definite matricesNorton, Peter George January 1964 (has links)
It is shown that the number of classes of nonisometric lattices on the space of rational ntuples is the same as the number of classes of n x n integral, symmetric, positive definite, unimodular matrices under integral congruence. A method is given to determine the number of classes of nonisometric lattices; this method is used to determine the number of classes for n↖ 16. A representative of each class of symmetric, integral, positive definite, unimodular 16x16 matrices is given. / Science, Faculty of / Mathematics, Department of / Graduate

7 
Group matricesIwata, William Takashi January 1965 (has links)
A new proof is given of Newman and Taussky's result: if A is a unimodular integral n x n matrix such that A′A is a circulant, then A = QC where Q is a generalized permutation matrix and C is a circulant. A similar result is proved for unimodular integral skew circulants.
Certain additional new results are obtained, the most interesting of which are: 1) Given any nonsingular group matrix A there exist unique real group matrices U and H such that U is orthogonal and H is positive definite and A = UH; 2) If A is any unimodular integral circulant, then integers k and s exist such that A′ = P(k)A and P(s)A is symmetric, where P is the companion matrix of the polynomial xⁿ1.
Finally, all the n x n positive definite integral and unimodular skew circulants are determined for values of n ≤ 6: they are shown to be trivial for n = 1,2,3 and are explicitly described for n = 4,5,6. / Science, Faculty of / Mathematics, Department of / Graduate

8 
Some combinatorial properties of matricesWales, David Bertram January 1962 (has links)
Three combinatorial problems in matrix theory are considered in this thesis.
In the first problem the structure of 01 matrices with permanent 1, 2, or 3, is analysed. It is shown that a 01 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 01 matrix with permanent p. The structure of fully indecomposable 01 matrices with permanent 1, 2, or 3, is then determined. It is found that the only fully indecomposable 01 matrix with permanent 1 is the 1square matrix (1); and fully indecomposable 01 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 01 matrices with permanent 3 is also described.
In the second problem relationships are considered between two 01 matrices given certain connections between their corresponding principal submatrices. The matrices considered are 01 nsquare 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 nsquare matrices A such that for any nsquare permutation matrix P there exists a doubly stochastic matrix B such that PA = AB. It is shown that for nonsingular matrices in G, the doubly stochastic matrix B must be a permutation matrix. The set of nonsingular 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 nsquare matrix with all entries 1, and α, β are scalars. / Science, Faculty of / Mathematics, Department of / Graduate

9 
Pairs of matrices with property L.Chow, Jihou January 1958 (has links)
Let A and B be nsquare 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 powerproducts 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

10 
Theory and applications of compound matricesThompson, Robert Charles January 1956 (has links)
If A is an nsquare matrix, the pth compound of A is a matrix whose entries are the pth 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 SylvesterFranke theorem. This theory is used to obtain an extremum property of elementary symmetric functions of the k largest (or smallest) eigenvalues of nonnegative 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

Page generated in 0.0776 seconds