In this thesis, employing the theory of matrix Lie groups, we develop gradient based flows for the problem of Simultaneous or Joint Diagonalization (JD) of a set of symmetric matrices. This problem has applications in many fields especially in the field of
Independent Component Analysis (ICA). We consider both orthogonal
and non-orthogonal JD. We view the JD problem as minimization of a
common quadric cost function on a matrix group. We derive gradient
based flows together with suitable discretizations for
minimization of this cost function on the
Riemannian manifolds of O(n) and GL(n).\\
We use the developed JD methods to introduce a new class of ICA
algorithms that sphere the data, however do not restrict the
subsequent search for the un-mixing matrix to orthogonal matrices.
These methods provide robust ICA algorithms in Gaussian noise by
making effective use of both second and higher order statistics