We say that a symmetric matrix K is quasi-definite if it has the form K = [ −E AT A F ] where E and F are symmetric positive definite matrices. Although such matrices are indefinite, we show that any symmetric permutation of a quasi-definite matrix yields a factorization LDLT . We apply this result to obtain a new approach for solving the symmetric indefinite systems arising in interior-point m...

We address the problem of learning a symmetric positive definite matrix. The central issue is to design parameter updates that preserve positive definiteness. Our updates are motivated with the von Neumann divergence. Rather than treating the most general case, we focus on two key applications that exemplify our methods: On-line learning with a simple square loss and finding a symmetric positiv...

We prove that the diagonally pivoted symmetric LR algorithm on a positive definite matrix is globally convergent. The “symmetric” or “Cholesky” LR iteration is a fairly old method of eigenreduction of a positive definite Hermitian matrix H. It reads H = H0 = R∗ 0R0 H1 = R0R 0 = R ∗ 1R1 .. (1) This process is linearly convergent [6], [5]. Recently, its singular value ’implicit’ equivalent R∗ k =...

In this paper, a sufficient and necessary condition for the matrix equations B AX = and , D XC = where , , n m n m B A × × ∈ ∈ R R , p n C × ∈R and , p n D × ∈ R to have a common symmetric positive semi-definite solution X is established, and if it exists, a representation of the solution set X S is given. An optimal approximation between a given matrix n n X × ∈ R ~ and the affine subspace X S...

We propose a new approximate factorization for solving linear systems with symmetric positive definite sparse matrices. In a nutshell the algorithm is to apply hierarchically block Gaussian elimination and additionally compress the fill-in. The systems that have efficient compression of the fill-in mostly arise from discretization of partial differential equations. We show that the resulting fa...

In many engineering applications that use tensor analysis, such as tensor imaging, the underlying tensors have the characteristic of being positive definite. It might therefore be more appropriate to use techniques specially adapted to such tensors. We will describe the geometry and calculus on the Riemannian symmetric space of positive-definite tensors. First, we will explain why the geometry,...

In this paper we consider domain decomposition methods for solving the radial basis function interpolation equations. There are three interwoven threads to the paper. The first thread provides good ways of setting up and solving smallto medium-sized radial basis function interpolation problems. These may occur as subproblems in a domain decomposition solution of a larger interpolation problem. ...