Factorizations of PSD Matrix Polynomials and their Smith Normal Forms
It is well-known, that any univariate polynomial matrix A over the
complex numbers that takes only positive semidefinite values on the real
line, can be factored as A=B^*B for a polynomial square matrix B.
This is a simultaneous generalization of both the fact that a univariate real
polynomial that is psd factors as a complex polynomial times its conjugate
and that a constant psd matrix A can be factored as B^*B
For real A, in general, one cannot choose B to be also a real square matrix.
However, if A is of size nxn, then a factorization A=B^tB exists, where
B is a real rectangular matrix of size (n+1)xn. We will see, how these
correspond to the factorizations of the Smith normal form of A, an
invariant not usually associated with symmetric matrices in their role
as quadratic forms. A consequence is, that the factorizations can
usually be easily counted, which in turn has an interesting application
to minimal length sums of squares of linear forms on varieties of
minimal degree.
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