Polynomial Matrix Spectral Factorization or Matrix Fejér–Riesz Theorem is a tool used to study the matrix decomposition of polynomial matrices. Polynomial matrices are widely studied in the fields of systems theory and control theory and have seen other uses relating to stable polynomials. In stability theory, Spectral Factorization has been used to find determinantal matrix representations for bivariate stable polynomials and real zero polynomials. Given a positive real trigonometric polynomial p ( t ) > 0 {\displaystyle p(t)>0} for all t ∈ R {\displaystyle t\in \mathbb {R} } , the Fejér–Riesz Theorem yields the factorization p ( t ) = q ( t ) q ¯ ( t ) {\displaystyle p(t)=q(t){\bar {q}}(t)} called the spectral factorization (or Wiener-Hopf factorization) of p ( t ) {\displaystyle p(t)} . Results of this form are generically referred to as Positivstellensatz. Likewise, the Polynomial Matrix Spectral Factorization provides a factorization for positive definite polynomial matrices. This decomposition also relates to the Cholesky decomposition for scalar matrices A = L L ∗ {\displaystyle A=LL^{*}} . This result was originally proven by Norbert Wiener in a more general context which was concerned with integrable matrix-valued functions that also had integrable log determinant. Because applications are often concerned with the polynomial restriction, simpler proofs and individual analysis exist focusing on this case. Weaker positivstellensatz conditions have been studied, specifically considering when the polynomial matrix has positive definite image on semi-algebraic subsets of the reals. Spectral factorization is used extensively in linear–quadratic–Gaussian control and many algorithms exist to calculate spectral factors. Some modern algorithms focus on the more general setting originally studied by Wiener while others have used Toeplitz matrix advances to speed up factor calculations.
Definition Consider the n × n {\displaystyle n\times n} polynomial matrix
P ( x ) = ∑ k = 0 N P k x k = [ p 11 ( x ) … p 1 n ( x ) ⋮ ⋱ ⋮ p n 1 ( x ) ⋯ p n n ( x ) ] , {\displaystyle P(x)=\sum _{k=0}^{N}P_{k}x^{k}={\begin{bmatrix}p_{11}(x)&\ldots &p_{1n}(x)\\\vdots &\ddots &\vdots \\p_{n1}(x)&\cdots &p_{nn}(x)\\\end{bmatrix}},}
where each element p i j ( x ) {\displaystyle p_{ij}(x)} is a complex polynomial of at most N {\displaystyle N} -degree. If P ( x ) {\displaystyle P(x)} is a positive semi-definite matrix for all x ∈ R {\displaystyle x\in \mathbb {R} } , then there exists an n × n {\displaystyle n\times n} polynomial matrix
Q ( x ) {\displaystyle Q(x)} with complex polynomials q i j ( x ) {\displaystyle q_{ij}(x)} such that
P ( x ) = Q ( x ) Q ∗ ( x ) , {\displaystyle P(x)=Q(x)Q^{*}(x),} where Q ∗ ( x ) {\displaystyle Q^{*}(x)} is the conjugate transpose. When the elements of Q ( x ) {\displaystyle Q(x)} are complex polynomials or complex coefficient rational functions then so are the elements of its conjugate transpose. We can furthermore find Q ( x ) {\displaystyle Q(x)} which is nonsingular on the lower half plane.
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