Lanczos method for large-scale quaternion singular value decomposition

In many color image processing and recognition applications, one of the most important targets is to compute the optimal low-rank approximations to color images, which can be reconstructed with a small number of dominant singular value decomposition (SVD) triplets of quaternion matrices. All existing methods are designed to compute all SVD triplets of quaternion matrices at first and then to select the necessary dominant ones for reconstruction. This way costs quite a lot of operational flops and CPU times to compute many superfluous SVD triplets. In this paper, we propose a Lanczos-based method of computing partial (several dominant) SVD triplets of the large-scale quaternion matrices. The partial bidiagonalization of large-scale quaternion matrices is derived by using the Lanczos iteration, and the reorthogonalization and thick-restart techniques are also utilized in the implementation. An algorithm is presented to compute the partial quaternion singular value decomposition. Numerical examples, including principal component analysis, color face recognition, video compression and color image completion, illustrate that the performance of the developed Lanczos-based method for low-rank quaternion approximation is better than that of the state-of-the-art methods.

     

Minimal ranks of some quaternion matrix expressions with applications

Suppose that p(X, Y) = A − BX − X^(∗)B^(∗) − CYC^(∗) and q(X, Y) = A − BX + X^(∗)B^(∗) − CYC^(∗) are quaternion matrix expressions, where A is persymmetric or perskew-symmetric. We in this paper derive the minimal rank formula of p(X, Y) with respect to pair of matrices X and Y = Y^(∗), and the minimal rank formula of q(X, Y) with respect to pair of matrices X and Y = −Y^(∗). As applications, we establish some necessary and sufficient conditions for the existence of the general (persymmetric or perskew-symmetric) solutions to some well-known linear quaternion matrix equations. The expressions are also given for the corresponding general solutions of the matrix equations when the solvability conditions are satisfied. At the same time, some useful consequences are also developed.     

Condensed Cramer rule for some restricted quaternion linear equations

In this paper, we derive an alternative more condensed Cramer rule for the unique solution of some restricted left and right systems of quaternion linear equations. The findings of this paper extend some known results in the literature.     

Determinantal representations of the generalized inverses A_{T,S}^{(2)} over the quaternion skew field with applications

We first present some determinantal representations of one {1,5}-inverse of a quaternion matrix within the framework of a theory of the row and column determinants. As applications, we show some new explicit expressions of generalized inverses $A_{r_{T_{1}, S_{1}}}^{( 2)}$ , $A_{l_{T_{2},S_{2}}}^{(2)}$ and $A_{_{( T_{1},T_{2}) , ( S_{1},S_{2}) }}^{( 2) }$ over the quaternion skew field. Finally, we give the representations of the unique solution to some restricted left and right systems of quaternionic linear equations. The findings of this paper extend some known results in the literature.     

Ranks of submatrices in (skew-)Hermitian solutions to a quaternion matrix equation

Assume that X and Y are Hermitian solutions to quaternion matrix equation AXA ^?+BYB ^?=C, which are partitioned into 2×2 block forms. We in this paper give the maximal and minimal ranks of submatrices in the Hermitian solutions X=X ^?, Y=Y ^?, and establish necessary and sufficient conditions for the submatrices to be zero, unique as well as independent. The findings of this paper widely extend the known results in the literature.     

New results on condensed Cramer’s rule for the general solution to some restricted quaternion matrix equations

In this paper, we derive some condensed Cramer’s rules for the general solution, the least squares solution and the least norm solution to some restricted quaternion matrix equations, respectively. The findings of this paper extend some known results in the literature.     

Bott-Duffin inverse over the quaternion skew field with applications

In this paper, we show some properties of the Bott-Duffin inverses $A_{r_{ ( L_{1} ) }}^{ ( -1 ) }$ and $A_{l_{ ( L_{2} ) }}^{ ( -1 ) }$ over the quaternion skew field. In particular, we establish the determinantal representations of these generalized inverses by the theory of the column and row determinants. Moreover, we derive some Cramer rules for the unique solution to some restricted linear quaternion equations. The findings of this paper extend some known results in the literature.     

Common nonnegative definite solutions of some classical matrix equations

A necessary and sufficient condition for the existence of a Hermitian nonnegative definite solution of system of matrix equations $$A_{1}X=C_{1},\qquad XB_{2}=C_{2}, \qquad A_{3}XA_{3}^{\ast}=C_{3},\qquad A_{4}XA_{4}^{\ast}=C_{4} $$ as well as a representation for this general nonnegative definite solution are derived. As particular cases, the corresponding results on some other systems are also derived.