In this thesis, we study the exact eigenvalue distribution of product of independent rectangular complex Gaussian matrices and also that of product of independent truncated Haar unitary matrices and inverses of truncated Haar unitary matrices. The eigenvalues of these random matrices form determinantal point processes on the complex plane. We also study the limiting expected empirical distribution of appropriately scaled eigenvalues of those matrices as the size of matrices go to infinity. We give the first example of a random matrix whose eigenvalues form a non-rotation invariant determinantal point process on the plane. The second theme of this thesis is infinite products of random matrices. We study the asymptotic behaviour of singular values and absolute values of eigenvalues of product of i .i .d matrices of fixed size, as the number of matrices in the product in-creases to infinity. In the special case of isotropic random matrices, We derive the asymptotic joint probability density of the singular values and also that of the absolute values of eigenvalues of product of right isotropic random matrices and show them to be equal. As a corollary of these results, we show probability that all the eigenvalues of product of certain i .i .d real random matrices of fixed size converges to one, as the number of matrices in the product increases to infinity.

During the last twenty years the theory of linear algebraic and high-order differential equation systems has been greatly researched. Two commonly used types of system description are the so-called matrix fraction description (MFD) and the Rosenbrock system matrix (RSM); these are defined by polynomial matrices in one indeterminate. Many of the system's physical properties are encoded as algebraic properties of these polynomial matrices. The theory is well developed and the structure of such systems is well understood. Analogues of these 1-D realisations can be set up for many dimensional systems resulting in polynomial matrices in many indeterminates. The scarcity of detailed algebraic results for such matrices has limited the understanding of such systems.

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Problems concerning transmission lines have been solved in the past by treating the line in terms of lumped parameters. Pioneering work was done by L. V. Bewley and S. Hayashi in the application of matrix theory to solve polyphase multiconductor distributed parameter transmission system problems. The availability of digital computers and the increasing complexity of power systems has renewed the interest in this field. With this in mind, a systematic procedure for handling complex transmission systems was evolved. Underlying the procedure is the significant concept of a complete system which defines how the parametric inductance, capacitance, leakance and resistance matrices must be formed and used. Also of significance is the use of connection matrices for handling transpositions and bonding, together with development of the manipulation of these matrices and the complex (Z) and (T) matrices. In the numerical procedure, methods were found to transform complex matrices into real matrices of twice the order and to determine the coefficients in the general solution systematically. The procedure was used to deal with phase asymmetry and mixed end boundary conditions. / Applied Science, Faculty of / Electrical and Computer Engineering, Department of / Graduate

The purpose of this thesis is to give a survey of the methods currently used to solve the numerical eigenvalue problem for real symmetric matrices. On the basis of the advantages and disadvantages inherent in the various methods, it is concluded that Householder's method is the best. Since the methods of Givens, Lanczos, and Householder use the Sturm sequence bisection algorithm as the final stage, a complete theoretical discussion of this process is included. Error bounds from a floating point error analysis (due to Ortega), for the Householder reduction are given. In addition, there is a complete error analysis for the bisection process. / Science, Faculty of / Mathematics, Department of / Graduate

In this thesis two problems concerning linear transformations on Mn, the algebra of n-square matrices over the complex numbers, are considered. The first is the determination of the structure of those transformations which map non-singular matrices to non-singular matrices; the second is the determination of the structure of those transformations which, for some positive integer r, preserve the sum of the r x r principal subdeterminants of each matrix. In what follows, we use E to denote this sum, and the phrase "direct product" to refer to transformations of the form T(A) = cUAV for all A in Mn or T(A) = cUA'V for all A in Mn where U, V are fixed members of Mn and c is a complex number. The main result of the thesis is that both non-singularity preservers and Er-preservers, if r ≥ 4, are direct products. The cases r=1,2,3 are discussed separately. If r=1, it is shown that E₁ preservers have no significant structure. If r=2, it is shown that there are two types of linear transformations which preserve E₂, and which are not direct products. Finally, it is shown that these counter examples do not generalize to the case r=3. These results and their proofs will also be found in a forthcoming paper by M. Marcus and JR. Purves in the Canadian Journal of Mathematics, entitled Linear Transformations of Algebras of Matrices: Invariance of the Elementary Symmetric Functions. / Science, Faculty of / Mathematics, Department of / Graduate

A number of potentials have been suggested as approximations to the 'true' potential function for the nuclei of a diatomic molecule. The relative merits of these potentials are discussed. Whenever possible the eigenfunctions and eigenvalues corresponding to these potentials are given. For the Morse potential the calculations of the eigenfunctions and eigenvalues are reproduced in detail. These eigenfunctions are used to derive general formulae for the radial parts of the dipole and quadrupole matrix elements. The expression for the dipole matrix element is [formula omitted] and for the quadrupole matrix element [formula omitted] The symbols are defined in sections 20, 21, and 22, The expression for M[subscript D] is in agreement with the one derived by Infeld and Hull while the expression for M[subscript Q] is a result which, so far as the author is aware, has not been published in the literature. / Science, Faculty of / Physics and Astronomy, Department of / Graduate

Existence problems (proving that a set is nonempty) abound in mathematics, so we look for generally applicable solutions (such as optimization techniques). To test and improve these techniques, we apply them to the Hadamard Conjecture (proving that Hadamard matrices exist in dimensions divisible by 4), which is a good example to study since Hadamard matrices have interesting applications (communication theory, quantum information theory, experiment design, etc.), are challenging to find, are easily distinguished from other matrices, are known to exist for many dimensions, etc.. In this thesis we study optimization algorithms (Exhaustive search, Hill Climbing, Metropolis, Gradient methods, generalizations thereof, etc.), improve their performance (when using a Graphical Processing Unit), and use them to attempt to find Hadamard matrices (real, and complex). Finally, we give an algorithm to prove non-trivial lower bounds on the Hamming distance between any given matrix with elements in {+1,-1} and the set of Hadamard matrices, then we use this algorithm to study matrices with similar properties to Hadamard matrices, but which are far away (with respect to the Hamming distance) from them.

Matricial norms, minimal matricial norms, vectorial norms and vectorial norms subordinate to matricial norms, which are respectively generalizations of matrix norms, minimal matrix norms, vector norms and vectorial norms subordinate to matrix norms, are defined and their various applications and properties are discussed. / Science, Faculty of / Mathematics, Department of / Graduate