Quadratic realizability of palindromic matrix polynomials: the real case

Let L = (L-1, L-2) be a list consisting of structural data for a matrix polynomial; here L-1 is a sublist consisting of powers of irreducible (monic) scalar polynomials over the field R, and L-2 is a sublist of nonnegative integers. For an arbitrary such L, we give easy-to-check necessary and sufficient conditions for L to be the list of elementary divisors and minimal indices of some real T-palindromic quadratic matrix polynomial. For a list L satisfying these conditions, we show how to explicitly build a real T-palindromic quadratic matrix polynomial having L as its structural data; that is, we provide a T-palindromic quadratic realization of L over R. A significant feature of our construction differentiates it from related work in the literature; the realizations constructed here are direct sums of blocks with low bandwidth, that transparently display the spectral and singular structural data in the original list L.