All Questions
31 questions
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What are the injective embeddings of R^d into the cone of (semi-) positive definite matrices of dimension d?
How can we characterize the set of all injective functions from $\mathbb{R}^d$ to the set of all symmetric positive definite matrices of dimension d?
- 109
0
votes
0
answers
49
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Conditions on symmetric $3 \times 3$ matrices to satisfy the convex equality for cofactor and determinant
Given any $3\times 3$ finite set of symmetric matrices $A_i$ and positive real $a_i$ such that $\sum_ia_i=1.$ Is there any equivalent condition to the existence of skew symmetric matrices $X_i$ such ...
1
vote
1
answer
206
views
Factorizing the doubly stochastic matrix where all entries are equal such that the factors are all convex combinations of few permutation matrices
Let $N_{n}=(1/n)_{i=1,j=1}^{n}$ be the $n\times n$-matrix where all the entries are equal.
Suppose $n>0$. Let $\delta_{n}$ be the least natural number such that $N_{n}$ can be factored as $N_{n}=A_{...
1
vote
1
answer
188
views
Is this $(\Bbb R^{n \times n})^n \to \Bbb R$ function convex?
Let $W := (W_1, W_2,\dots, W_n)$, where $W_i \in \Bbb R^{n \times n}$. Let $x$ be a constant vector. Is the following function convex?
$$f(W) := x^TW_1^TW_2^T \cdots W_n^TW_n \cdots W_2W_1x $$
- 11
0
votes
1
answer
765
views
Prove the optimal solution to maximizing nuclear norm with constraints is attained at corner points of feasible region
The nuclear norm (trace norm) of a matrix $X \in \Bbb R^{m \times n}$ is defined as
$$\|X\|_* := \sum_{i=1}^{\min(m,n)} \sigma_i(X)$$
where $\sigma_i(X)$ are the singular values of $X$.
The ...
- 43
11
votes
1
answer
1k
views
A square root inequality for symmetric matrices?
In this post all my matrices will be $\mathbb R^{N\times N}$ symmetric positive semi-definite (psd), but I am also interested in the Hermitian case. In particular the square root $A^{\frac 12}$ of a ...
- 5,380
4
votes
1
answer
151
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Mapping inclusion theorem for the numerical range
We denote the numerical range of a complex square matrix $A \in \mathbb{C}^{n\times n}$ by $W(A)$.
Let $A \in \mathbb{C}^{n\times n}$ and let $f: \mathbb{C} \to \mathbb{C}$ be, say, an entire ...
- 12.6k
3
votes
1
answer
228
views
Convexity of the matrix mapping $X^{-2}$
Let $X$ be a positive semidefinite matrix. Is the mapping $X\to X^{-2}$ convex?
Update: or is $Tr[X^{-2} K]$ convex for PSD $X$ and $K$?
- 181
11
votes
2
answers
559
views
Convex hull of the Stiefel manifold with non-negativity constraints
Consider the Stiefel manifold
$$\mathrm{St}(n,k) :=\{X \in \mathbb{R}^{n\times k} : X^TX = I_k\},$$
where $I_k$ is the $k$-dimensional identity matrix. It is well known that
$$\mathrm{conv} \left( ...
- 1,991
1
vote
1
answer
313
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Nonlinear low-rank approximation - corrected
I would like to state that this is related to a past question of mine which contained errors and now appears in the corrected form, with the erroneous one deleted and closed.
In my research of linear ...
- 620
4
votes
0
answers
70
views
"Singularly convex" cones of matrices
The ambient space if ${\bf M}_n({\mathbb R})$.
Let us begin with facts.
1- The cone of positive semi-definite symmetric matrices is convex. 2- It is a little subtler that the cone $K^+$ of matrices ...
- 52.3k
3
votes
1
answer
332
views
Positive semi-definite in the limit
Consider the $n\times n$ matrix $F$ defined by the following expression
$$
F=A-\varepsilon B
$$
where $A$ is a constant matrix such that $a_{ij}=a>0$ for all $i,j$ and where $B$ is a symmetric ...
- 379
0
votes
0
answers
40
views
convex representation of a combinatorial constraint
I have an optimization problem with a weird constraint as follows. Is it possible to express it in some ways that have convex properties:
matrix $\mathbf{X}$ is either
$[1 \ 0 \ 0 \ 0 \ 0\\
\ 0 \ 0 ...
2
votes
1
answer
167
views
Carathéodory's theorem for $SO(3)$?
Let $Q \in \operatorname{Conv} SO(3)$.
Is there a way to retrieve an explicit representation of $Q$ as convex combination $Q=\sum_{k=1}^{r}{\lambda_{i}R_{i}},
R_{i} \in SO(3)$?
An approximation ...
- 6,962
3
votes
2
answers
290
views
Eigenspace of convex combination of two idempotent matrices
Let $H_1,H_2\in\mathbb{Q}^{n\times n}$ be idempotent and symmetric matrices. For any $0<\mu<\frac{1}{2}$, consider the matrix
$$H_\mu:=\mu H_1+(1-\mu)H_2.$$
I'm looking for a description of $\...
- 268
0
votes
2
answers
3k
views
Convexity of the Frobenius norm of the product of two matrices
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 ...
- 291
5
votes
1
answer
704
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$ ...
- 6,962
3
votes
1
answer
1k
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 ...
- 6,962
0
votes
1
answer
203
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}
\mathbf{C}=\left(\mathbf{I}*\frac{1}{\...
- 1,421
2
votes
1
answer
665
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 ...
- 6,962
0
votes
0
answers
237
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:
$M_0(A,B)=A^{\frac{1}{2}}(A^{-\frac{1}{2}}BA^{-...
- 155
7
votes
2
answers
315
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
...
- 421
0
votes
1
answer
180
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 ...
- 6,962
3
votes
1
answer
135
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 ...
- 1,352
4
votes
1
answer
1k
views
Generalizing inequality relating Euclidean distance & Frobenius norm to Bregman divergences 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 ...
- 607
3
votes
1
answer
525
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)$.
...
user2529
5
votes
2
answers
1k
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 ...
user2529
1
vote
0
answers
227
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 ...
- 519
7
votes
3
answers
6k
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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 semi-...
- 181
2
votes
0
answers
240
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 f(\omega,t)$...
- 52.3k
5
votes
1
answer
796
views
Orthogonal similarity of matrices
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 ...
- 52.3k