science

Linear matrix inequality

Linear matrix inequality is a science topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Linear matrix inequality rather than just read about it. In short: In convex optimization, a linear matrix inequality (LMI) is an expression of the form LMI ⁡ ( y ) := A 0 + y 1 A 1 + y 2 A 2 + ⋯ + y m A m ⪰ 0 {\displaystyle \operatorname {LMI} (y):=A_{0}+y_{1}A_{1}+y_{2}A_{2}+\cdots +y_{m}A_{m}\succeq 0\,} where y = [ y i , i = 1 , … , m ] {\displaystyle y=[y_{i}\,,~i\!=\!1,\dots ,m]} is a real vector, A 0 , A 1 , A 2 , … , A m {\displaystyle A_{0},A_{1},A_{2},\dots ,A_{m}} are n…

Key takeaways

Linear matrix inequality belongs to science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Linear matrix inequality to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Linear matrix inequality from memory before moving on to harder problems.

Reference excerpt

In convex optimization, a linear matrix inequality (LMI) is an expression of the form

LMI ⁡ ( y ) := A 0 + y 1 A 1 + y 2 A 2 + ⋯ + y m A m ⪰ 0 {\displaystyle \operatorname {LMI} (y):=A_{0}+y_{1}A_{1}+y_{2}A_{2}+\cdots +y_{m}A_{m}\succeq 0\,}

where

y = [ y i , i = 1 , … , m ] {\displaystyle y=[y_{i}\,,~i\!=\!1,\dots ,m]} is a real vector,

A 0 , A 1 , A 2 , … , A m {\displaystyle A_{0},A_{1},A_{2},\dots ,A_{m}} are n × n {\displaystyle n\times n} symmetric matrices S n {\displaystyle \mathbb {S} ^{n}} ,

B ⪰ 0 {\displaystyle B\succeq 0} is a generalized inequality meaning B {\displaystyle B} is a positive semidefinite matrix belonging to the positive semidefinite cone S + {\displaystyle \mathbb {S} _{+}} in the subspace of symmetric matrices S {\displaystyle \mathbb {S} } . This linear matrix inequality specifies a convex constraint on y {\displaystyle y} .

Applications There are efficient numerical methods to determine whether an LMI is feasible (e.g., whether there exists a vector y such that LMI(y) ≥ 0), or to solve a convex optimization problem with LMI constraints. Many optimization problems in control theory, system identification and signal processing can be formulated using LMIs. Also LMIs find application in Polynomial Sum-Of-Squares. The prototypical primal and dual semidefinite program is a minimization of a real linear function respectively subject to the primal and dual convex cones governing this LMI.

Solving LMIs A major breakthrough in convex optimization was the introduction of interior-point methods. These methods were developed in a series of papers and became of true interest in the context of LMI problems in the work of Yurii Nesterov and Arkadi Nemirovski.

See also Semidefinite programming Spectrahedron Finsler's lemma

References Y. Nesterov and A. Nemirovsky, Interior Point Polynomial Methods in Convex Programming. SIAM, 1994.

External links S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory (book in pdf) C. Scherer and S. Weiland, Linear Matrix Inequalities in Control

Worked examples

Example 1 — a first encounter with Linear matrix inequality

Start with the simplest possible case. Write down what Linear matrix inequality claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Linear matrix inequality before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Linear matrix inequality ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Linear matrix inequality

In research: Linear matrix inequality appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Linear matrix inequality in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Linear matrix inequality is common in secondary-school and first-year university syllabi. It links to neighbouring topics Convex optimization, so understanding it makes those chapters shorter.
In everyday life: Look for Linear matrix inequality outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

Ask Teacher Smith questions about this articleOpens your AI tutor with a question about “Linear matrix inequality” →

Affiliate

Preply — study more efficiently by working with a personal tutor. 50% off.

How to study Linear matrix inequality in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Linear matrix inequality means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Linear matrix inequality out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Linear matrix inequality in simple terms?

In convex optimization, a linear matrix inequality (LMI) is an expression of the form LMI ⁡ ( y ) := A 0 + y 1 A 1 + y 2 A 2 + ⋯ + y m A m ⪰ 0 {\displaystyle \operatorname {LMI} (y):=A_{0}+y_{1}A_{1}+y_{2}A_{2}+\cdots +y_{m}A_{m}\succeq 0\,} where y = [ y i , i = 1 , … , m ] {\displaystyle y=[y_{i}…

Why does Linear matrix inequality matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Linear matrix inequality?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Linear matrix inequality.