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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Applying an affine transformation from the matrix to vector representations of spectahedra

On page 9, Parrilo wrote$^\color{magenta}{\star}$ that the vector representation of a spectrahedron $$ S = \left\{ (x_1, \dots, x_m) \in {\Bbb R}^m : A_0 + \sum_{i=1}^m A_i x_i \succeq 0 \right\} $$ ...
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Show that $\{x \mid A_0 + \sum x_i A_i \succcurlyeq 0 \}$ is convex

Let $A_0, A_1,\dots,A_m$ be symmetric matrices. Let $x \in \mathbb R^m$ and define $$A(x) := A_0 + \sum_{i=1}^m x_i A_i$$ Show that the set $C := \{x \mid A(x) \text{ is positive semidefinite} \}$ ...
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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 = \...
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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... ...
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Is the set of positive definite matrices with trace one an open subset of hermitian matrices?

Is the set of positive definite matrices with trace one an open subset of hermitian matrices? I know the set of positive definite matrices is open, but I don't know how to prove that the trace one ...
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Calculation involving determinant of a matrix

Suppose I have the following Toeplitz symmetric matrix \begin{align} M=\begin{bmatrix} 1 & c & c & x \\ c & 1 & c & c \\ c & c & 1 & c \\ x & c & c & 1 ...
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Is there a way to identify singular points in a spectrahedron without finding the entire set?

I'm a physicist working in a non-linear optimization problem that reduces to a semidefinite program (SDP). In the simple low dimensional cases we workout in the research we could identify singular ...
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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 = \left\{\textbf{W} \in \mathbb{R}^{d \times d} \mid \...
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Finding the dual of the feasible set of a linear matrix inequality (LMI)

I am stuck at problem 3 of this homework assignment about semidefinite programming and linear matrix inequalities (LMIs). Given a set $\mathcal{S} \subseteq \mathbb{R}^n$ that strictly contains the ...
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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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Convexity of the set $C = \{x \in \Bbb R^n \mid x^T Ax \leq b\}$ where $A \in S^n$, $A \succeq 0$ and $b \geq 0$

I have this question in my homework and neither I nor my colleagues can solve it. Show that the set $$C = \{x \in \Bbb R^n \mid x^T Ax \leq b\}$$ where $A$ is symmetric and positive semidefinite ...
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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 elliptope ...
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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 & 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 + ...
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