# Questions tagged [linear-matrix-inequalities]

Linear Matrix Inequalities (LMIs)

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### Solving Problem: LMIs and block matrices

I have been reading through this paper (https://ieeexplore.ieee.org/document/7995739) where I am stuck with this particular LMI. If you are familiar with control theory, the author is trying to find ...

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### Linear matrix inequality

I have the following linear matrix inequality:
$$F^T P + PF < 0,$$
where $P$ is a positive definite matrix and $F$ is a matrix with appropriate dimension.
Let $Q$ be a positive definite matrix ...

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### Positive definiteness of a Matrix

$K>0\in \mathbb{R}^{n\times n}$, $P>0 \in \mathbb{R}^{n\times n}$ are diagonal positive definite matrices. And $R\geq 0\in \mathbb{R}^{m\times m}$ is positive semi-definite matrix. Let $B\in \...

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### Why two matrix sets defined by such LMI are equivalent?

Suppose
$\{(x_1,x_2) : x_1^2+x_2^2 = 1\}$ the unit circle.
Consider two sets defined by a quadratic constraint and LMI:
$$\{Y\in R^{2\times 2}: \begin{bmatrix}x_1 & x_2 \end{bmatrix}\...

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### Is a spectrahedron's boundary almost always “smooth”?

A spectrahedron is a convex set defined by a linear matrix inequality (LMI).
Is the boundary of such a set almost always smooth?
By "smooth" I mean that it admits a tangent hyperplane at any point ...

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### Explicit formula for an LMI solution

Suppose we have a linear matrix inequality (aka LMI aka spectahedron aka linear matrix pencil):
$$A_{0}+x_{1}A_{1}+x_{2}A_{2}+\ldots+x_{m}A_{m} \succeq 0.$$
(The notation $X \succeq Y$ means that $X-...

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### Convexity of a specific semialgebraic set

I have an engineering problem that may be solved using semidefinite programming. I would like to know whether a given set is convex.
Let $m \in \mathbb{R}^+$ be a positive real scalar, $l \in \mathbb{...