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.