A Block-Jacobi Algorithm for Non-Symmetric Joint Diagonalization of Matrices

This paper studies the problem of Non-symmetric Joint Diagonalization (NsJD) of matrices, namely, jointly diagonalizing a set of complex matrices by one matrix multiplication from the left and one multiplication with possibly another matrix from the right. In order to avoid ambiguities in the solutions, these two matrices are restricted to lie in the complex oblique manifold. We derive a necessary and sufficient condition for the uniqueness of solutions, and characterize the Hessian of global minimizers of the off-norm cost function, which is typically used to measure joint diagonality. By exploiting existing results on Jacobi algorithms, we develop a block-Jacobi method that guarantees local convergence to a pair of joint diagonalizers at a quadratic rate. The performance of the proposed algorithm is investigated by numerical experiments.