Standard linear algebra concepts (linear equation, inverses, QR factorization, least squares, eigenvalue decomposition, singular value decomposition, etc.) naturally extended to matrices with elements from associative non-commutative algebra. Examples include matrices of quaternions, matrices whose elements are again square matrices or even multi-dimensional arrays like tensors. We analyze definitions which hold in this more general setting and analyze existing and provide many new interesting algorithms.

An essential ingredient of many algorithms is a corresponding Givens rotation, which is easily defined for quaternions or block matrices. While many basic algorithms are simple generalizations of their standard counterparts, finer details of modern algorithms over real or complex field are difficult to implement in the case of non-commutativity.

We will investigate problems and find possible resolutions for efficient algorithms in this more complex setting. Applications of new algorithms to image processing, in particular ultrasound tomography, using quaternions or tensors will also be analyzed.

##### Project objectives
1. Inverses of structured block arrowhead and diagonal-plus rank-k matrices. Application to Rayleigh quotient block method for such matrices.
2. Analysis of methods for computing the eigenvalues and singular values of matrices of quaternions.
3. Methods for the computation of  eigenvalues and singular values of Hermitian matrices of quaternions and block matrices and pairs of such matrices.
4. Methods for computing the eigenvalues of general matrices of quaternions and block matrices, in particular: analysis of an ideal shift in a non-commutative case, power method and inverse iterations, and Rayleigh quotient method.
5. Methods for computing eigenvalues and singular values of structured (arrow head and diagonal-plus-rank-k) matrices of quaternions or block matrices. Applications to optimal damping.
6. Methods for calculating the roots of  polynomials of quaternions by reducing the problem to finding eigenvalues of structured matrices of quaternions.
7. Method for solving least squares problem and total least squares problem for matrices of quaternions and block matrices.
8. Application of quaternions to ultrasonic tomography.