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Let X = Lp(Ω,μ) (1

In this article, we give the density functions of the singular quaternion normal matrix and the singular quaternion Wishart matrix. Furthermore, we also give the density functions of certain singular quaternion β-matrix and the singular quaternion F-matrix in terms of the density function of the singular quaternion Wishart matrix and hypergeometric functions of quaternion matrix argument.

We define zonal polynomials of quaternion matrix argument and deduce some important formulae of zonal polynomials and hypergeometric functions of quaternion matrix argument. As an application, we give the distributions of the largest and smallest eigenvalues of a quaternion central Wishart matrix $W\sim\mathbb{Q}W(n,\Sigma)$, respectively., Comment: 22 pages. Communications in Statistics - Theory and Methods (appear)

Let L be a subspace of ${\bf R}^n$ and P L be the orthogonal projection of R n onto L . Then for the n × n matrix A , the generalized Bott-Duffin (B-D) inverse $A^{(+)}_{(L)}$ is given by $A^{(+)}_{(L)}=P_L(AP_L+I-P_L)^+$ . In this article we prove that $A^{(+)}_{(L)}=(P_LAP_L)^+$ iff $AL\cap L^\perp =0$ . This result extends the concept so-called the " L -s.p.d" matrix proposed by Chen in his article "The Generalized Bott-Duffin Inverse and its Applications". In the rest part of the article, the perturbation analysis of $A^{(+)}_{(L)}$ and the least squares solution of the systems $Ax+B^*y=b, Bx=d$ are established under certain small perturbation of A .