# Tagged Questions

I have the following function for two matrices ${\bf A}$ and ${\bf B}$: $f({\bf A}, {\bf B}) = \| {\bf Y - XAB} \|_F^2 = trace\{({\bf Y - XAB)}^T({\bf Y - XAB)}\}$ where matrices ${\bf X}_{n \times ... 1answer 244 views ### What it is the volume of the unit ball section of the cone of positive definite matrices? Let$PD_{n}$be the cone of positive definite$n \times n$real matrices and let$B$be the unit sphere in$n \times n$dimensions. What is the volume of$PD_{n} \cap B$? EDIT: Let's assume that$B$... 1answer 118 views ### Is this function of a matrix convex? Let$\mathcal{N}_{n}$be the set of symmetric nonnegative irreducible matrices. For a matrix$A \in \mathcal{N}_{n}$let$v^{A}$be its Perron vector, normalized so that$||v^{A}||_{2}=1$. Define the ... 1answer 105 views ### Eigenvalues of a given parametrized matrix. Let$\mathbf{A}$and$\mathbf{B}$be two complex rank-one$N\times N$positive semi-definite matrices. Let the matrix$\mathbf{C}be defined as \begin{align} ... 1answer 205 views ### Covering the cone of positive semidefinite matrices by intervals Is it possible to cover the cone of positive semidefinite matrices by a finite/countable/interesting family of closed intervals of matrices? How about a general convex cone? For the finite case the ... 0answers 180 views ### Geometric Mean of Positive Matrices Hello all, My question regards the geometric mean (GM) of two positive matrices. The definition of the GM for two positive matrices(A,B)$is given by: ... 2answers 267 views ### Duality between extremal points and extremal maps Suppose I have a convex set$C\subset\mathbb{R}^n$such that$0\in C$and every Cauchy sequence in$C$converges in$C$, but$C$need not be bounded. (Actually I want unbounded$C$). Consider the set ... 1answer 159 views ### (probably simple) optimization question Suppose you have a concave function defined over a non-polyhedral convex cone and you are interested in the infimum. What would be standard approaches to tackle the question? (The cone is actually PSD ... 1answer 116 views ### Mapping a subset of semi-definite matrices through arcsinus Hi I am meeting a problem concerning semi-definite positive matrices, and I have no clue concerning them, the classical approaches I know have not given any result, maybe people used to manipulating ... 1answer 319 views ### Is it possible to extend this inequality about Euclidean distance &Frobenius norm to more general Bregman divergence such as relative entropy & von Neumann divergence? Motivation- A Special Case Supposing$A,B\in\mathbb{S}^{m\times m}$are symmetric positive semi-definite (SPD) matrices and$\mathbf{x}\in\mathbb{R}^m$is a unit vector where$\|\mathbf{x}\|=1$, we ... 0answers 294 views ### What is the “positive part” of the unit ball in$M_n(R)$? In${\bf M}_n(\mathbb R)$, let us consider the usual operator norm $$\|A\|=\sup\frac{\|Ax\|}{\|x\|},$$ where$\|x\|$is the Euclidian norm. The closed unit ball$B$is the set of contractions (in the ... 1answer 243 views ### Linear and Isometric Automorphism Groups of the PSD Cone Let$S_+$be the cone of psd matrices ($n\times n$real symmetric positive semidefinite matrices). This cone is a metric space induced from the inner product$\langle A,B\rangle = tr (AB)=tr(BA)$. ... 2answers 409 views ### Is a solution of a linear system of semidefinite matrices a convex combination of rank 1 solutions? The cone of symmetric positive semidefinite$n\times n$matrices is the convex hull of rank$1$matrices. That is, every symmetric positive semidefinite matrix is a convex combination of rank 1 ... 0answers 179 views ### Joint Convexity of Spectral functions of several matrices$\{A_1 \ldots A_K \}$is a set of matrices in$\mathbb{R}^{m \times n}$. Let$f (A_1,\ldots,A_K)$be a function of the singular values of all matrices. For e.g.,$f$is just summation of singular ... 3answers 2k views ### Minimize trace of inverse of convex combination of matrices. Hello! (First question--please forgive me if its unclear.) I am interested in efficient/approximate optimization techniques for minimizing a norm of a convex combination of symmetric, positive ... 0answers 174 views ### Radon transform and Log-concavity This question is related to (but different from) that of Darsh Ranjan. Is there a characterization of the functions$f:\mathbb R^n\rightarrow\mathbb R_{\ge0}$whose Radon transform$\hat ...
Given $M\in M_n({\mathbb R})$ and $\ell\in{0,\ldots,n-1}$, we define $$d_\ell(M)=\sum_{j=1}^nm_{j,j+\ell},$$ where the indices are understood mod $n$. In particular, $d_0$ is the trace operator. Let ...