Quadratic realizability of palindromic matrix polynomials
Date of Original Version
Let L=(L 1 ,L 2 ) be a list consisting of a sublist L 1 of powers of irreducible (monic) scalar polynomials over an algebraically closed field F, and a sublist L 2 of nonnegative integers. For an arbitrary such list L, we give easily verifiable necessary and sufficient conditions for L to be the list of elementary divisors and minimal indices of some T-palindromic quadratic matrix polynomial with entries in the field F. For L satisfying these conditions, we show how to explicitly construct a T-palindromic quadratic matrix polynomial having L as its structural data; that is, we provide a T-palindromic quadratic realization of L. Our construction of T-palindromic realizations is accomplished by taking a direct sum of low bandwidth T-palindromic blocks, closely resembling the Kronecker canonical form of matrix pencils. An immediate consequence of our in-depth study of the structure of T-palindromic quadratic polynomials is that all even grade T-palindromic matrix polynomials have a T-palindromic strong quadratification. Finally, using a particular Möbius transformation, we show how all of our results can be easily extended to quadratic matrix polynomials with T-even structure.
Linear Algebra and Its Applications
De Terán, Fernando, Froilán M. Dopico, D. Steven Mackey, and Vasilije Perović. "Quadratic realizability of palindromic matrix polynomials." Linear Algebra and Its Applications 567, (2019): 202-262. doi:10.1016/j.laa.2019.01.003.