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Questions tagged [spectrahedra]

A spectrahedron is a convex set given by the intersection of an affine space with the convex positive semidefinite cone.

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In spectrahedra, are minimal rank points always extreme points?

Consider a matrix $A$ in a spectrahedron $S$ such that $$\mbox{rank}(A) \leq \mbox{rank}(B)$$ for all $B\in S$ and assume that at least one matrix $C \in S$ we have that $\mbox{rank}(A) < \mbox{...
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Geometric center of convex set of positive semidefinite matrices

Consider the set of $d\times d$ matrices that are positive semidefinite and have unit trace. This is a convex set, $S$. Is it possible to think of a geometric center of this set? The criterion for ...
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What is the surface area of the 3-dimensional elliptope?

The $n$-elliptope is defined as the set of $n$-by-$n$ correlation matrices; that is, the set of $n$-by-$n$ symmetric positive-definite matrices with ones on the diagonal. Such matrices are ...
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What is the formula for projection onto spectraplex?

A spectraplex (special case of spectrahedron) is the set of all positive semi-definite matrices whose trace is equal to one. Formally, let $$ S=\{\textbf{W} \in \mathbb{R}^{d \times d} \mid \textbf{W} ...
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Does cone associated with PSD matrix always convex?

Is it true that PSD-cone always convex? (If not, please provide an example). If this is the case, then set of PSD matrices always convex or it could happen that such set might not be convex for some ...
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Every convex polyhedron is a spectrahedron

I'm trying to show that convex polyhedra are special cases of spectrahedra. This was left as an exercise to the reader in a convex optimization text that I'm reading. I'm not sure how standard the ...
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Why is the feasible set of solutions to an SDP a spectrahedron?

A spectrahedron is the set $$ S = \left\lbrace (x_1,\cdots,x_m)\in \mathbb{R}^m \quad|\quad A_0+ A_ix_i \succeq 0 \quad i\in [m] \right\rbrace$$ for some given symmetric matrices $A_0, A_1,\cdots,...
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Is the set $S = \{x \in [0, \infty)^2 \mid x_1 x_2^2 \leq 1\}$ convex?

We have the function $\displaystyle{y=f(x_1, x_2)=x_1\cdot x_2^2}$ and the set $S=\{x\in [0, \infty)^2 \mid f(x_1, x_2)\leq 1\}$. I want to check if the set is convex. $$$$ Let $x=(x_1, x_2) , y=...
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What is the volume of the $3$-dimensional elliptope?

My question Compute the following double integral analytically $$\int_{-1}^1 \int_{-1}^1 2 \sqrt{x^2 y^2 - x^2 - y^2 + 1} \,\, \mathrm{d} x \mathrm{d} y$$ Background The $3$-dimensional ...
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What can we say about the form of the solution set $\{B : \mbox{tr} (AB) < 0\}$?

I have two $n \times n$ matrices $A$ and $B$, where $B$ is symmetric and p.s.d and $A$ is symmetric , rank $2$ and its two dominant eigenvalues have different signs. Considering the following ...
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How to plot the PSD cone in MATLAB

Does anybody know how I can plot in MATLAB the cone of positive semidefinite matrices as shown in the figure below? Thanks. PSD cone
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A spectrahedron is a quadric cone when matrices in LMI are in $\mathbb{R}^{2\times 2}$

I am watching a lecture (just at the beginning around 0:50-0:57). The note says when $n=2$, $\mathcal{S}$ is a quadric cone; however, it seems that the professor says "a quadratic cone". On ...
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Solution set of an LMI is convex

I was going through Boyd and Vandenberghe's Convex Optimization book. There they mentioned (at page number $38$) that the solution set of a linear matrix inequality (LMI) is convex. $$A(x)=x_1A_1+\...