A structure-preserving one-sided Jacobi method for computing the SVD of a quaternion matrix

Abstract(#br)In this paper, we propose a structure-preserving one-sided cyclic Jacobi method for computing the singular value decomposition of a quaternion matrix. In our method, the columns of the quaternion matrix are orthogonalized in pairs by using a sequence of orthogonal JRS-symplectic Jacobi matrices to its real counterpart. We establish the quadratic convergence of our method specially. We also give some numerical examples to illustrate the effectiveness of the proposed method

Quantum conductance problems and the Jacobi ensemble

In one dimensional transport problems the scattering matrix $S$ is decomposed into a block structure corresponding to reflection and transmission matrices at the two ends. For $S$ a random unitary matrix, the singular value probability distribution function of these blocks is calculated. The same is done when $S$ is constrained to be symmetric, or to be self dual quaternion real, or when $S$ has real elements, or has real quaternion elements. Three methods are used: metric forms; a variant of the Ingham-Seigel matrix integral; and a theorem specifying the Jacobi random matrix ensemble in terms of Wishart distributed matrices.Comment: 10 page

From Random Matrices to Stochastic Operators

We propose that classical random matrix models are properly viewed as finite difference schemes for stochastic differential operators. Three particular stochastic operators commonly arise, each associated with a familiar class of local eigenvalue behavior. The stochastic Airy operator displays soft edge behavior, associated with the Airy kernel. The stochastic Bessel operator displays hard edge behavior, associated with the Bessel kernel. The article concludes with suggestions for a stochastic sine operator, which would display bulk behavior, associated with the sine kernel.Comment: 41 pages, 5 figures. Submitted to Journal of Statistical Physics. Changes in this revision: recomputed Monte Carlo simulations, added reference [19], fit into margins, performed minor editin

Simultaneous diagonalisation of the covariance and complementary covariance matrices in quaternion widely linear signal processing

Recent developments in quaternion-valued widely linear processing have established that the exploitation of complete second-order statistics requires consideration of both the standard covariance and the three complementary covariance matrices. Although such matrices have a tremendous amount of structure and their decomposition is a powerful tool in a variety of applications, the non-commutative nature of the quaternion product has been prohibitive to the development of quaternion uncorrelating transforms. To this end, we introduce novel techniques for a simultaneous decomposition of the covariance and complementary covariance matrices in the quaternion domain, whereby the quaternion version of the Takagi factorisation is explored to diagonalise symmetric quaternion-valued matrices. This gives new insights into the quaternion uncorrelating transform (QUT) and forms a basis for the proposed quaternion approximate uncorrelating transform (QAUT) which simultaneously diagonalises all four covariance matrices associated with improper quaternion signals. The effectiveness of the proposed uncorrelating transforms is validated by simulations on both synthetic and real-world quaternion-valued signals.Comment: 41 pages, single column, 10 figure

Relaxed 2-D Principal Component Analysis by LpL_p Norm for Face Recognition

A relaxed two dimensional principal component analysis (R2DPCA) approach is proposed for face recognition. Different to the 2DPCA, 2DPCA-$L_1$ and G2DPCA, the R2DPCA utilizes the label information (if known) of training samples to calculate a relaxation vector and presents a weight to each subset of training data. A new relaxed scatter matrix is defined and the computed projection axes are able to increase the accuracy of face recognition. The optimal $L_p$-norms are selected in a reasonable range. Numerical experiments on practical face databased indicate that the R2DPCA has high generalization ability and can achieve a higher recognition rate than state-of-the-art methods.Comment: 19 pages, 11 figure

The generalized Cartan decomposition for classical random matrix ensembles

We present a completed classification of the classical random matrix ensembles: Hermite (Gaussian), Laguerre (Wishart), Jacobi (MANOVA) and Circular by introducing the concept of the generalized Cartan decomposition and the double coset space. Previous authors associate a symmetric space $G/K$ with a random matrix density on the double coset structure $K\backslash G/K$. However this is incomplete. Complete coverage requires the double coset structure $A = K_1\backslash G/K_2$, where $G/K_1$ and $G/K_2$ are two symmetric spaces. Furthermore, we show how the matrix factorization obtained by the generalized Cartan decomposition $G = K_1AK_2$ plays a crucial role in sampling algorithms and the derivation of the joint probability density of $A$.Comment: 26 page