# Questions tagged [linear-matrix-inequalities]

Linear Matrix Inequalities (LMIs)

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### Inequality for normed power n, m

Let $ B (H) $ indicate the set of all bounded linear operators on a complex separable Hilbert space $ H $.
Let $ A \in B(H) $, where $ A $ is a positive semi-definite operator in $ H $ (i.e. $ \langle ...

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### Inverting a linear system of inequalities

Suppose I have a set of inequalities $l \leq Ax \leq r$ for vectors $l,r$ and matrix $A$. I am trying to bound $x$ using a function of $l,r: f(l) \leq x \leq g(r)$ for some $f,g$. When can we invert ...

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### A matrix inequality

Suppose we have three semi-definite positive matrices $A,B,C$ with $A+B=I$.
Do we have the following inequality
$$
Tr[(CAC)^{1+s}B]\leq Tr [(CBC)^s CAC]
$$
holds for any $0<s<1$?
This is true ...

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### Matrix inversion inequality

Suppose $A, B, C\in\mathbb{R}^{n\times n}$ are all symmetric positive definite matrices, and they satisfy the inequality $A \succeq B + C$. Assume also that all of the three matrices are bounded, i.e.,...

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### Non-conservatively bound a positive definite matrix

Suppose a matrix $A_k$ is a symmetric positive definite real matrix. As $k$ increases, $A_k$ may change but will always be symmetric positive definite. What I want to do is to have an operation $f:\...

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### Approximation bounds for matrix multiplication

$\DeclareMathOperator{\op}{\mathrm{op}}$Since matrix multiplication is continuous, I expect that if $A_n\to A\in \mathrm{Mat}_{d\times d}(\mathbb{R})$ for the operator-norm and if $x_n\to x$ in $\...

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### Minimum operator that exceeds others (in a PSD, linear matrix inequality, sense)

Given a collection of $n$ matrices $A_i$, we could ask for the $B$ such that:
$$\textrm{Minimize: }\quad \textrm{Tr}[B]$$
$$\textrm{Such that: }\forall_i\, B \succeq A_i$$
Here $\succeq$ is in the ...

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### Reference request for linear matrix inequality with PSD matrices

In literature, people say a spectrahedron is the following set
$$\left\{x \in \mathbb{R}^d : x_1 A_1 + \cdots + x_d A_d \geq B \right\}$$
where $\geq$ is in the positive semidefinite sense. Is there a ...

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### Quantum inspired matrix inequality

While mimicking the union bound in quantum systems, we land on the following conjecture but don't know how to prove this. Given any complex-valued $n\times m$ matrix $A$. A sub-matrix of $A$ is ...

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### Sufficient conditions for a system of linear inequalities to admit a solution

I am looking for sufficient conditions such that a system of linear inequalities of the type $A x >0$ admits a non-negative solution $x \in \mathbb{R}^n_+$. I know a few properties of the $m \times ...

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### A Matrix Inequality for positive definite matrices

Let $X$ and $Y$ be positive semi-definite self-adjoint complex matrices of same finite order. The, is it true that $|X-Y|\leq X+Y$ where for any matrix $A$, $|A|$ is defined to be $|A|:=(A^*A)^{\frac{...

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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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vote

1
answer

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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 ...

4
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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-...

5
votes

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295
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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{...