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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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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 parametrized by their $n(n-1)/2$ upper off-diagonal…
Semiclassical
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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 elliptope is the spectrahedron defined as…
Rodrigo de Azevedo
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Solution set of an LMI is convex

I was going through Boyd & Vandenberghe's Convex Optimization. On page 38, the authors mentioned that the solution set of a linear matrix inequality (LMI) is convex. $$ A(x) := x_1 A_1 + \dots + x_n A_n \preceq B $$ where $A_1, \dots, A_n, B \in…
Rajat
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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 = \left\{\textbf{W} \in \mathbb{R}^{d \times d} \mid \textbf{W} \succeq 0, \text{Tr}(\textbf{W})=1…
user494522
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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 definitions and notation in this book are, so I'll…
zxmkn
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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 geometric center is an element $\rho$ such that if…
user1936752
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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 $x=(x_1, x_2) , y=(y_1, y_2)\in S$, then $x_1\cdot x_2^2\leq 1$ and…
Mary Star
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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 inequality: $$\{B : \mbox{tr} (AB)<0\}$$ Meaning that we…
Bob
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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
PGriffin
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Extreme points of elliptopes

Let $K$ be set of all positive matrices $A \in M_3(\mathbb{C})$ such that all the diagonal entries are $1$. Observe that $K$ is a convex set. An element $A\in K$ is extreme if it is not interior point of a line segment in $K$ i.e. if $\lambda\in…
DeltaEpsilon
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Finding the left root of $-x^2 + 2x + 1$ via convex optimization

Can we formulate as a linear matrix inequality (LMI) one of the regions in which a concave quadratic polynomial is negative (without solving its roots)? Let the quadratic polynomial $f \in {\Bbb R} [x]$ be concave and with positive discriminant…
Morad
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Vertices of spectrahedron in Lovasz SDP

Currently, I am studying Lovasz's semidefinite program (SDP) to calculate $\alpha(G)$ for perfect graphs. And I encountered a paper$^\color{magenta}{\star}$ by Silva & Tunçel. A vertex of a convex set is an extreme point of that set whose normal…
Risss
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How do I interpret this picture of a spectrahedron?

On the Wikipedia page on spectrahedron, it shows the following picture of a spectrahedron (I avoid using the term image since it seems to have special meaning in this context) I understand everything on the wikipedia page and have a rudimentary…
Rufus
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Linear matrix inequality and convex epigraph

In example 3.4 of Boyd & Vandenberghe's Convex Optimization, function $f : \mathbb{R}^n \times \mathbb{S}^n \to \mathbb{R}$, defined as $$f(x, Y) := x^T Y^{-1}x$$ is convex on $\mathrm{dom} f = \mathbb{R}^n \times \mathbb{S}_{++}^n$, where…
user21
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Why is the boundary of spectrahedra “more pointy” at matrices of lower rank?

In the following expository article about spectrahedra, it is established informally that the boundary of spectrahedra is “more pointy” at matrices of lower rank. Cynthia Vinzant, What is a... Spectrahedron?, Notices of the AMS, Volume 61, Number…
titowed
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