In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a form

Homogeneous polynomial

In mathematics, a homogeneous polynomial is a polynomial whose monomials with nonzero coefficients all have thesame total degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial...

 (i.e. a homogeneous polynomial) h(x) of degree 2m in the real n-dimensional vector x is sum of squares of forms (SOS) if and only if there exist forms of degree m such that


Explicit sufficient conditions for a form to be SOS have been found. However every real nonnegative form can be approximated as closely as desired (in the -norm of its coefficient vector) by a sequence of forms that are SOS.

Square matricial representation (SMR)

To establish whether a form h(x) is SOS amounts to solving a convex optimization problem. Indeed, any h(x) can be written as


where is a vector containing a base for the forms of degree m in x (such as all monomials of degree m in x), the prime ′ denotes the transpose

Transpose

In linear algebra, the transpose of a matrix A is another matrix AT created by any one of the following equivalent actions:...

, H is any symmetric matrix satisfying


and is a linear parameterization of the linear space

Vector space

A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

The dimension of the vector is given by


whereas the dimension of the vector is given by


Then, h(x) is SOS if and only if there exists a vector such that


meaning that the matrix is positive-semidefinite. This is a linear matrix inequality (LMI) feasibility test, which is a convex optimization problem. The expression was introduced in [1] with the name square matricial representation (SMR) in order to establish whether a form is SOS via an LMI. This representation is also known as Gram matrix (see [2] and references therein).

Examples

• Consider the form of degree 4 in two variables . We have
  Since there exists α such that , namely , it follows that h(x) is SOS.
  
• Consider the form of degree 4 in three variables . We have
  Since for , it follows that h(x) is SOS.
  

Matrix SOS

A matrix form F(x) (i.e., a matrix whose entries are forms) of dimension r and degree 2m in the real n-dimensional vector x is SOS if and only if there exist matrix forms of degree m such that

Matrix SMR

To establish whether a matrix form F(x) is SOS amounts to solving a convex optimization problem. Indeed, similarly to the scalar case any F(x) can be written according to the SMR as


where is the Kronecker product

Kronecker product

In mathematics, the Kronecker product, denoted by ⊗, is an operation on two matrices of arbitrary size resulting in a block matrix. It gives the matrix of the tensor product with respect to a standard choice of basis. The Kronecker product should not be confused with the usual matrix...

of matrices, H is any symmetric matrix satisfying


and is a linear parameterization of the linear space


The dimension of the vector is given by


Then, F(x) is SOS if and only if there exists a vector such that the following LMI holds:


The expression was introduced in [3] in order to establish whether a matrix form is SOS via an LMI.