Spectral theory and solving infinite matrices
Linear algebra is based on the well known equation of a line, y=mx+b, where m is the slope of the line and b is its y-intercept. If b=0, the equation becomes y=mx, or mx=y. This simple equation becomes slightly more complex when we look at systems of equations as represented by matrices. In the matrix equation Ax=b, the matrix A is officially called the operator, acting on the input x to produce the output b. However, for very large or infinite matrices, linear algebra is often inadequate to solve the associated problems. The ‘Spectra, fredholm properties and stable approximation of infinite matrices’ (Infinitematrices) project set out to develop ways to derive spectral information associated with infinite matrices, develop innovative ways of solving infinite linear systems and apply the results to specific operators important to acoustic and electromagnetic (EM) wave theory and quantum mechanics. The researchers studied the spectra of infinite matrices and analysed the asymptotic behaviour of the matrix entries to determine the essential spectra. They also studied finite principal submatrices to determine information about other parts of the spectra. This work led to derived new upper bounds on the spectra that complemented the lower bounds derived previously. The researchers also addressed methods of solving infinite linear systems (described by infinite matrices) using innovative truncation techniques. They sought to overcome the limitations of the conventional method, ‘cutting’ finite square submatrices from the original and solving the smaller ones. A rigorous study enabled them to guarantee the convergence that often eludes conventional truncation methods by avoiding certain matrix sizes and by using rectangular instead of square submatrices. Finally, the researchers demonstrated not only the general use of these mathematical techniques in solving infinite matrices but also applied them to specific operators relevant to acoustic and EM wave theory as well as quantum mechanics. Thus, the EU-funded Infinitematrices project led to significant advances in spectral theory and stable approximation of infinite matrices that should be applicable to numerous fields in the engineering and natural sciences.
