Symmetric cone

Key takeaways

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

Reference excerpt

In mathematics, symmetric cones, sometimes called domains of positivity, are open convex self-dual cones in Euclidean space which have a transitive group of symmetries, i.e. invertible operators that take the cone onto itself. By the Koecher–Vinberg theorem these correspond to the cone of squares in finite-dimensional real Euclidean Jordan algebras, originally studied and classified by Jordan, von Neumann & Wigner (1934). The tube domain associated with a symmetric cone is a noncompact Hermitian symmetric space of tube type. All the algebraic and geometric structures associated with the symmetric space can be expressed naturally in terms of the Jordan algebra. The other irreducible Hermitian symmetric spaces of noncompact type correspond to Siegel domains of the second kind. These can be described in terms of more complicated structures called Jordan triple systems, which generalize Jordan algebras without identity.

Definitions A convex cone C in a finite-dimensional real inner product space V is a convex set invariant under multiplication by positive scalars. It spans the subspace C – C and the largest subspace it contains is C ∩ (−C). It spans the whole space if and only if it contains a basis. Since the convex hull of the basis is a polytope with non-empty interior, this happens if and only if C has non-empty interior. The interior in this case is also a convex cone. Moreover, an open convex cone coincides with the interior of its closure, since any interior point in the closure must lie in the interior of some polytope in the original cone. A convex cone is said to be proper if its closure, also a cone, contains no subspaces. Let C be an open convex cone. Its dual is defined as

C ∗ = { X : ( X , Y ) > 0 f o r Y ∈ C ¯ } . {\displaystyle \displaystyle {C^{*}=\{X:(X,Y)>0\,\,\mathrm {for} \,\,Y\in {\overline {C}}\}.}}

It is also an open convex cone and C** = C. An open convex cone C is said to be self-dual if C* = C. It is necessarily proper, since it does not contain 0, so cannot contain both X and −X. The automorphism group of an open convex cone is defined by

A u t C = { g ∈ G L ( V ) | g C = C } . {\displaystyle \displaystyle {\mathrm {Aut} \,C=\{g\in \mathrm {GL} (V)|gC=C\}.}}

Clearly g lies in Aut C if and only if g takes the closure of C onto itself. So Aut C is a closed subgroup of GL(V) and hence a Lie group. Moreover, Aut C* = (Aut C)*, where g* is the adjoint of g. C is said to be homogeneous if Aut C acts transitively on C. The open convex cone C is called a symmetric cone if it is self-dual and homogeneous.

Group theoretic properties If C is a symmetric cone, then Aut C is closed under taking adjoints. The identity component Aut0 C acts transitively on C. The stabilizers of points are maximal compact subgroups, all conjugate, and exhaust the maximal compact subgroups of Aut C. In Aut0 C the stabilizers of points are maximal compact subgroups, all conjugate, and exhaust the maximal compact subgroups of Aut0 C. The maximal compact subgroups of Aut0 C are connected. The component group of Aut C is isomorphic to the component group of a maximal compact subgroup and therefore finite. Aut C ∩ O(V) and Aut0 C ∩ O(V) are maximal compact subgroups in Aut C and Aut0 C. C is naturally a Riemannian symmetric space isomorphic to G / K where G = Aut0 C. The Cartan involution is defined by σ(g)=(g*)−1, so that K = G ∩ O(V).

Spectral decomposition in a Euclidean Jordan algebra

In their classic paper, Jordan, von Neumann & Wigner (1934) studied and completely classified a class of finite-dimensional Jordan algebras, that are now called either Euclidean Jordan algebras or formally real Jordan algebras.

Definition Let E be a finite-dimensional real vector space with a symmetric bilinear product operation

E × E → E , a , b ↦ a b = b a , {\displaystyle \displaystyle {E\times E\rightarrow E,\,\,\,a,b\mapsto ab=ba,}}

with an identity element 1 such that a1 = a for a in A and a real inner product (a,b) for which the multiplication operators L(a) defined by L(a)b = ab on E are self-adjoint and satisfy the Jordan relation

L ( a ) L ( a 2 ) = L ( a 2 ) L ( a ) . {\displaystyle \displaystyle {L(a)L(a^{2})=L(a^{2})L(a).}}

… excerpt ends here. Continue reading the full article.

Illustrations

Symmetric cone: Max Koecher pioneered the use of Jordan algebras in studying symmetric spaces — Max Koecher pioneered the use of Jordan algebras in studying symmetric spaces

Worked examples

Example 1 — a first encounter with Symmetric cone

Start with the simplest possible case. Write down what Symmetric cone claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, 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 Symmetric cone 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 Symmetric cone 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 Symmetric cone

In research: Symmetric cone appears in mathematics 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 Symmetric cone 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: Symmetric cone is common in secondary-school and first-year university syllabi. It links to neighbouring topics Convex geometry, Lie algebras, Lie groups, so understanding it makes those chapters shorter.
In everyday life: Look for Symmetric cone 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.

How to study Symmetric cone in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Symmetric cone 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 Symmetric cone out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Symmetric cone in simple terms?

In mathematics, symmetric cones, sometimes called domains of positivity, are open convex self-dual cones in Euclidean space which have a transitive group of symmetries, i.e. invertible operators that take the cone onto itself. By the Koecher–Vinberg theorem these correspond to the cone of squares i…

Why does Symmetric cone matter?

Because it connects several mathematics 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 Symmetric cone?

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 Symmetric cone.