Polynomial matrix spectral factorization

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 determinental matrix representations for bivariate stable polynomials and real zero polynomials. A key tool used to study these is a matrix factorization known as either the Polynomial Matrix Spectral Factorization or the Matrix Fejer–Riesz Theorem.
Given a univariate positive polynomial formula_1, a polynomial which takes on non-negative values for any real input formula_2, the Fejer–Riesz Theorem yields the polynomial spectral factorization formula_3. Results of this form are generically referred to as Positivstellensatz. Considering positive definiteness as the matrix analogue of positivity, Polynomial Matrix Spectral Factorization provides a similar factorization for polynomial matrices which have positive definite range. This decomposition also relates to the Cholesky decomposition for scalar matrices formula_4. This result was originally proven by 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. Many publications recently have focused on streamlining proofs for these related results. This article roughly follows the recent proof method of Lasha Ephremidze which relies only on elementary linear algebra and complex analysis.
Spectral Factorization is used extensively in linear–quadratic–Gaussian control. Because of this application there have been many algorithms to calculate spectral factors. Some modern algorithms focus on the more general setting originally studied by Wiener. In the formula_5 case the problem is known as polynomial spectral factorization, or Fejer-Riesz Theorem, and has many classical algorithms. Some modern algorithms have used Toeplitz matrix advances to speed up factor calculations.
Statement.
Let formula_6be a polynomial matrix where each entry formula_7 is a complex coefficient polynomial of degree at most formula_8. Suppose that for almost all formula_9 we have formula_10 is a positive definite hermitian matrix. Then there exists a polynomial matrix formula_11 such that formula_12 for all formula_9. We can furthermore find formula_11 which is nonsingular on the lower half plane.
Extension to complex inputs.
Note that if formula_15then formula_16. When formula_17 is a complex coefficient polynomial or complex coefficient rational function we have formula_18is also a polynomial or rational function respectively. For formula_9 we haveformula_20 This because of the following observation: Since the entries of formula_21 and formula_10 are complex polynomials which agree on the real line, they are in fact the same polynomials. We can conclude they in fact agree for all complex inputs.
Rational spectral factorization.
Let formula_1 be a rational polynomial function where formula_24 for almost all formula_9. Then there exists rational formula_26 with formula_3 where formula_26 has no poles or zeroes in the lower half plane. This decomposition is unique up to multiplication by complex scalars of norm formula_29. This is related to the statement of the Polynomial Matrix Spectral Factorization theorem restricted to the formula_5 case.
To prove existence write formula_31where formula_32. Letting formula_33, we can conclude that formula_34 is real and positive. Dividing out by formula_35 we reduce to the monic case. The numerator and denominator have distinct sets of roots, so all real roots which show up in either must have even multiplicity (to prevent a sign change locally). We can divide out these real roots to reduce to the case where formula_1 has only complex roots and poles. By hypothesis we have formula_37. Since all of the formula_38are complex (and hence not fixed points of conjugation) they both come in conjugate pairs. For each conjugate pair, pick the zero or pole in the upper half plane and accumulate these to obtain formula_26. The uniqueness result follows in a standard fashion.
Cholesky decomposition.
The inspiration for this result is a factorization which characterizes positive definite matrices.
Decomposition for scalar matrices.
Given any positive definite scalar matrix formula_40, the Cholesky decomposition allows us to write formula_41 where formula_42 is a lower triangular matrix. If we don't restrict to lower triangular matrices we can consider all factorizations of the form formula_43. It is not hard to check that all factorizations are achieved by looking at the orbit of formula_42 under right multiplication by a unitary matrix, formula_45.
To obtain the lower triangular decomposition we induct by splitting off the first row and first column:formula_46Solving these in terms of formula_47 we getformula_48formula_49formula_50
Since formula_40 is positive definite we have formula_52is a positive real number, so it has a square root. The last condition from induction since the right hand side is the Schur complement of formula_40, which is itself positive definite.
Decomposition for rational matrices.
Now consider formula_6where the formula_7are complex rational functions and formula_10 is positive definite hermitian for almost all real formula_2. Then by the symmetric Gaussian elimination we performed above, all we need to show is there exists a rational formula_58 such that formula_59for real formula_2, which follows from our rational spectral factorization. Once we have that then we can solve for formula_61. Since the Schur complement is positive definite for the real formula_2 away from the poles and the Schur complement is a rational polynomial matrix we can induct to find formula_63.
It is not hard to check that we in fact get formula_64where formula_65 is a rational polynomial matrix with no poles in the lower half plane.
Extension to polynomial decompositions.
To prove the existence of polynomial matrix spectral factorization, we begin with the rational polynomial matrix Cholesky Decomposition and modify it to remove lower half plane singularities. Namely given formula_6with each entry formula_7 a complex coefficient polynomial we have rational polynomial matrix formula_65 with formula_64for real formula_2, where formula_65 has no lower half plane poles. Given a rational polynomial matrix formula_72 which is unitary valued for real formula_2, there exists another decomposition,formula_74.
Removing lower half-plane singularities.
If formula_75 then there exists a scalar unitary matrix formula_76such that formula_77. This implies formula_78has first column vanish at formula_79. To remove the singularity at formula_79 we multiply byformula_81formula_82has determinant with one less zero (by multiplicity) at a, without introducing any poles in the lower half plane of any of the entries.
Extend analyticity to all of C.
After modifications, the decomposition formula_83satisfies formula_11 is analytic and invertible on the lower half plane. To extend analyticity to the upper half plane we need this key observation: Given a rational matrix formula_11 who is analytic in the lower half plane and nonsingular in the lower half plane, we have formula_86 is analytic and nonsingular in the lower half plane. The analyticity follows from the adjugate matrix formula (since both the entries of formula_11 and formula_88are analytic on the lower half plane). The nonsingularity follows from formula_89which can only have zeroes at places where formula_90 had poles. The determinant of a rational polynomial matrix can only have poles where its entries have poles, so formula_90 has no poles in the lower half plane.
From our observation in Extension to Complex Inputs, we have formula_92for all complex numbers. This implies formula_93. Since formula_86 is analytic on the lower half plane, formula_11 is analytic on the upper half plane. Finally if formula_11 has a pole on the real line then formula_97has the same pole on the real line which contradicts the fact formula_10 has no poles on the real line (it is analytic everywhere by hypothesis).
The above shows that if formula_11 is analytic and invertible on the lower half plane indeed formula_11 is analytic everywhere and hence a polynomial matrix.
Uniqueness.
Given two polynomial matrix decompositions which are invertible on the lower half planeformula_101 we rearrange to formula_102. Since formula_103 is analytic on the lower half plane and nonsingular, formula_104is a rational polynomial matrix which is analytic and invertible on the lower half plane. Then by the same argument as above we have formula_104 is in fact a polynomial matrix which is unitary for all real formula_2. This means that if formula_107 is the formula_108th row of formula_11then formula_110. For real formula_2 this is a sum of non-negative polynomials which sums to a constant, implying each of the summands are in fact constant polynomials. Then formula_112where formula_76 is a scalar unitary matrix.
Example.
Consider formula_114. Then through symmetric Gaussian elimination we get the rational decomposition formula_115. This decomposition has no poles in the upper half plane. However the determinant is formula_116, so we need to modify our decomposition to get rid of the singularity at formula_117. First we multiply by a scalar unitary matrix to make a column vanish at formula_108. Consider formula_119. Then we have a new candidate for our decompositionformula_120. Now the first column vanishes at
formula_108, so we multiply through (on the right) by formula_122 to obtain formula_123 Notice formula_124. This is our desired decomposition formula_125 with no singularities in the lower half plane.