On certain matrix algebras related to quasi-Toeplitz matrices

Abstract

AbstractLet $$A_\alpha $$ A α be the semi-infinite tridiagonal matrix having subdiagonal and superdiagonal unit entries, $$(A_\alpha )_{11}=\alpha $$ ( A α ) 11 = α , where $$\alpha \in \mathbb C$$ α ∈ C , and zero elsewhere. A basis $$\{P_0,P_1,P_2,\ldots \}$$ { P 0 , P 1 , P 2 , … } of the linear space $$\mathcal {P}_\alpha $$ P α spanned by the powers of $$A_\alpha $$ A α is determined, where $$P_0=I$$ P 0 = I , $$P_n=T_n+H_n$$ P n = T n + H n , $$T_n$$ T n is the symmetric Toeplitz matrix having ones in the nth super- and sub-diagonal, zeros elsewhere, and $$H_n$$ H n is the Hankel matrix with first row $$[\theta \alpha ^{n-2}, \theta \alpha ^{n-3}, \ldots , \theta , \alpha , 0, \ldots ]$$ [ θ α n - 2 , θ α n - 3 , … , θ , α , 0 , … ] , where $$\theta =\alpha ^2-1$$ θ = α 2 - 1 . The set $$\mathcal {P}_\alpha $$ P α is an algebra, and for $$\alpha \in \{-1,0,1\}$$ α ∈ { - 1 , 0 , 1 } , $$H_n$$ H n has only one nonzero anti-diagonal. This fact is exploited to provide a better representation of symmetric quasi-Toeplitz matrices $$\mathcal{Q}\mathcal{T}_S$$ Q T S , where, instead of representing a generic matrix $$A\in \mathcal{Q}\mathcal{T}_S$$ A ∈ Q T S as $$A=T+K$$ A = T + K , where T is Toeplitz and K is compact, it is represented as $$A=P+H$$ A = P + H , where $$P\in \mathcal {P}_\alpha $$ P ∈ P α and H is compact. It is shown experimentally that the matrix arithmetic obtained this way is much more effective than that implemented in the toolbox of Numer. Algo. 81(2):741–769, 2019.