In mathematics, a form (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
Contents
- Square matricial representation SMR
- Examples
- Matrix SOS
- Matrix SMR
- Noncommutative polynomial SOS
- References
Every form that is SOS is also a positive polynomial, and although the converse is not always true, Hilbert proved that for n = 2, m = 1 or n = 3 and 2m = 4 a form is SOS if and only if is positive. The same is also valid for the analog problem on positive symmetric forms.
Although not every form can be represented as SOS, explicit sufficient conditions for a form to be SOS have been found. Moreover, every real nonnegative form can be approximated as closely as desired (in the
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
and
The dimension of the vector
whereas the dimension of the vector
Then, h(x) is SOS if and only if there exists a vector
meaning that the matrix
Examples
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
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
and
The dimension of the vector
Then, F(x) is SOS if and only if there exists a vector
The expression
Noncommutative polynomial SOS
Consider the free algebra R⟨X⟩ generated by the n noncommuting letters X = (X1,...,Xn) and equipped with the involution T, such that T fixes R and X1,...,Xn and reverse words formed by X1,...,Xn. By analogy with the commutative case, the noncommutative symmetric polynomials f are the noncommutative polynomials of the form f=fT. When any real matrix of any dimension r x r is evaluated at a symmetric noncommutative polynomial f results in a positive semi-definite matrix, f is said to be matrix-positive.
A noncommutative polynomial is SOS if there exists noncommutative polynomials
Surprisingly, in the noncommutative scenario a noncommutative polynomial is SoS if and only if is matrix-positive. Moreover, there exist algorithms available to decompose matrix-positive polynomials in sum of squares of noncommutative polynomials.