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Many problems in applied mathematics and physics are formulated most naturally in terms of matrices, and can be solved by computing functions of these matrices. For example, in quantum mechanics, the coherent dynamics of physical systems is described by the matrix exponential of their Hamiltonian. In state of the art experiments, one can now observe such unitary evolution of many-body systems, which is of fundamental interest in the study of many-body quantum phenomena. On the other hand the theoretical simulation of such non-equilibrium many-body dynamics is very challenging. In this thesis, we develop a symbolic approach to matrix functions and quantum dynamics based on a novel algebraic structure we identify for sets of walks on graphs. We begin by establishing the graph theoretic equivalent to the fundamental theorem of arithmetic: all the walks on any finite digraph uniquely factorise into products of prime elements. These are the simple paths and simple cycles, walks forbidden from visiting any vertex more than once. We give an algorithm that efficiently factorises individual walks and obtain a recursive formula to factorise sets of walks. This yields a universal continued fraction representation for the formal series of all walks on digraphs. It only involves simple paths and simple cycles and is thus called a path-sum. In the second part, we recast matrix functions into path-sums. We present explicit results for a matrix raised to a complex power, the matrix exponential, matrix inverse, and matrix logarithm. We introduce generalised matrix powers which extend desirable properties of the Drazin inverse to all powers of a matrix. In the third part, we derive an intermediary form of path-sum, called walk-sum, relying solely on physical considerations. Walk-sum describes the dynamics of a quantum system as resulting from the coherent superposition of its histories, a discrete analogue to the Feynman path-integrals. Using walk-sum we simulate the dynamics of quantum random walks and of Rydberg-excited Mott insulators. Using path-sum, we demonstrate many-body Anderson localisation in an interacting disordered spin system. We give two observable signatures of this phenomenon: localisation of the system magnetisation and of the linear magnetic response function. Lastly we return to the study of sets of walks. We show that one can construct as many representations of series of walks as there are ways to define a walk product such that the factorisation of a walk always exist and is unique. Illustrating this result we briefly present three further methods to evaluate functions of matrices. Regardless of the method used, we show that graphs are uniquely characterised, up to an isomorphism, by the prime walks they sustain.

In this thesis, we present efficient deterministic algorithms for polynomial matrix computation problems, including the computation of order basis, minimal kernel basis, matrix inverse, column basis, unimodular completion, determinant, Hermite normal form, rank and rank profile for matrices of univariate polynomials over a field. The algorithm for kernel basis computation also immediately provides an efficient deterministic algorithm for solving linear systems. The algorithm for column basis also gives efficient deterministic algorithms for computing matrix GCDs, column reduced forms, and Popov normal forms for matrices of any dimension and any rank. We reduce all these problems to polynomial matrix multiplications. The computational costs of our algorithms are then similar to the costs of multiplying matrices, whose dimensions match the input matrix dimensions in the original problems, and whose degrees equal the average column degrees of the original input matrices in most cases. The use of the average column degrees instead of the commonly used matrix degrees, or equivalently the maximum column degrees, makes our computational costs more precise and tighter. In addition, the shifted minimal bases computed by our algorithms are more general than the standard minimal bases.

In this thesis, we present efficient deterministic algorithms for polynomial matrix computation problems, including the computation of order basis, minimal kernel basis, matrix inverse, column basis, unimodular completion, determinant, Hermite normal form, rank and rank profile for matrices of univariate polynomials over a field. The algorithm for kernel basis computation also immediately provides an efficient deterministic algorithm for solving linear systems. The algorithm for column basis also gives efficient deterministic algorithms for computing matrix GCDs, column reduced forms, and Popov normal forms for matrices of any dimension and any rank. We reduce all these problems to polynomial matrix multiplications. The computational costs of our algorithms are then similar to the costs of multiplying matrices, whose dimensions match the input matrix dimensions in the original problems, and whose degrees equal the average column degrees of the original input matrices in most cases. The use of the average column degrees instead of the commonly used matrix degrees, or equivalently the maximum column degrees, makes our computational costs more precise and tighter. In addition, the shifted minimal bases computed by our algorithms are more general than the standard minimal bases.

In Markov chain models in finance and healthcare a transition matrix over a certain time interval is needed but only a transition matrix over a longer time interval may be available. The problem arises of determining a stochastic $p$th root of astochastic matrix (the given transition matrix). By exploiting the theory of functions of matrices, we develop results on the existence and characterization of stochastic $p$th roots. Our contributions include characterization of when a real matrix hasa real $p$th root, a classification of $p$th roots of a possibly singular matrix,a sufficient condition for a $p$th root of a stochastic matrix to have unit row sums,and the identification of two classes of stochastic matrices that have stochastic $p$th roots for all $p$. We also delineate a wide variety of possible configurationsas regards existence, nature (primary or nonprimary), and number of stochastic roots,and develop a necessary condition for existence of a stochastic root in terms of the spectrum of the given matrix. On the computational side, we emphasize finding an approximate stochastic root: perturb the principal root $A^{1/p}$ or the principal logarithm $\log(A)$ to the nearest stochastic matrix or the nearest intensity matrix, respectively, if they are not valid ones;minimize the residual $\normF{X^p-A}$ over all stochastic matrices $X$ and also over stochastic matrices that are primary functions of $A$. For the first two nearness problems, the global minimizers are found in the Frobenius norm. For the last two nonlinear programming problems, we derive explicit formulae for the gradient and Hessian of the objective function $\normF{X^p-A}^2$ and investigate Newton's method, a spectral projected gradient method (SPGM) and the sequential quadratic programming method to solve the problem as well as various matrices to start the iteration. Numerical experiments show that SPGM starting with the perturbed $A^{1/p}$to minimize $\normF{X^p-A}$ over all stochastic matrices is method of choice.Finally, a new algorithm is developed for computing arbitrary real powers $A^\a$ of a matrix $A\in\mathbb{C}^{n\times n}$. The algorithm starts with a Schur decomposition,takes $k$ square roots of the triangular factor $T$, evaluates an $[m/m]$ Pad\'e approximant of $(1-x)^\a$ at $I - T^$, and squares the result $k$ times. The parameters $k$ and $m$ are chosen to minimize the cost subject to achieving double precision accuracy in the evaluation of the Pad\'e approximant, making use of a result that bounds the error in the matrix Pad\'e approximant by the error in the scalar Pad\'e approximant with argument the norm of the matrix. The Pad\'e approximant is evaluated from the continued fraction representation in bottom-up fashion, which is shown to be numerically stable. In the squaring phase the diagonal and first superdiagonal are computed from explicit formulae for $T^$, yielding increased accuracy. Since the basic algorithm is designed for $\a\in(-1,1)$, a criterion for reducing an arbitrary real $\a$ to this range is developed, making use of bounds for the condition number of the $A^\a$ problem. How best to compute $A^k$ for a negative integer $k$ is also investigated. In numerical experiments the new algorithm is found to be superior in accuracy and stability to several alternatives,including the use of an eigendecomposition, a method based on the Schur--Parlett\alg\ with our new algorithm applied to the diagonal blocks and approaches based on the formula $A^\a = \exp(\a\log(A))$.