1 Tensor Sylvester matrices and information matrices of multiple stationary processes

By André Klein, Department of Quantitaive Economics, University of Amsterdam, Roetersstraat 11, 1018 WB Amsterdam, The Netherlands.

Consider the matrix polynomials A(z) and B(z) given by

A(z) = ∑_{j=0}^{p} A_{j}z^{j}

and

B(z) = ∑_{j=0}^{q} B_{j}z^{j},

where A_{0} ≡ B_{0} ≡ I_{n}. Gohberg and Lerer [1] study the resultant property of the tensor Sylvester

matrix S^{⊗}(−B, A) ≜ S(−B⊗I_{n}, I_{n}⊗A) or S^{⊗}(−B, A) = ( \begin{matrix}(−I_{n}) ⊗ I_{n} & (−B_{1}) ⊗ I_{n} & ⋯ & (−B_{q}) ⊗ I_{n} & 0_{n^{2}×n^{2}} & ⋯ & 0_{n^{2}×n^{2}} \\ 0_{n^{2}×n^{2}} & ... & ... & & ... & ... & ⋮ \\ ⋮ & ... & ... & ... & & ... & 0_{n^{2}×n^{2}} \\ 0_{n^{2}×n^{2}} & ⋯ & 0_{n^{2}×n^{2}} & (−I_{n}) ⊗ I_{n} & (−B_{1}) ⊗ I_{n} & ⋯ & (−B_{q}) ⊗ I_{n} \\ I_{n} ⊗ I_{n} & I_{n} ⊗ A_{1} & ⋯ & I_{n} ⊗ A_{p} & 0_{n^{2}×n^{2}} & ⋯ & 0_{n^{2}×n^{2}} \\ 0_{n^{2}×n^{2}} & ... & ... & & ... & ... & ⋮ \\ ⋮ & ... & ... & ... & & ... & 0_{n^{2}×n^{2}} \\ 0_{n^{2}×n^{2}} & ⋯ & 0_{n^{2}×n^{2}} & I_{n} ⊗ I_{n} & I_{n} ⊗ A_{1} & ⋯ & I_{n} ⊗ A_{p} \end{matrix} ).

In [1] it is proved that the matrix polynomials A(z) and B(z) have at least one common eigenvalue if and only if detS^{⊗}(−B, A) = 0 or when the matrix S^{⊗}(−B, A) is singular. In other words, the tensor Sylvester matrix S^{⊗}(−B, A) becomes singular if and only if the scalar polynomials det A(z) = 0 and det B(z) = 0 have at least one common root. Consequently, it is a multiple resultant. In [2], this property is extended to the Fisher information matrix of a stationary vector autoregressive and moving average process, VARMA process. The purpose of this talk consists of displaying a representation of the Fisher information matrix of a stationary VARMAX process in terms of tensor Sylvester matrices, the X stands for exogenous or control variable. The VARMAX process is of common use in stochastic systems and control.