Some algebraic methods for analysing matrix continued fractions (Q6619683)

Let \(A_k, B_k, C_k \in \mathcal{M}_n(\mathbb{C}) \) be matrices \((k = 0, 1, 2,\dots)\). The expression\N\[\NA_0+B_1\bigg(A_1 +B_2\Big(A_2+B_3\big(A_3+\dots \big)^{-1} C_3 \Big)^{-1} C_2 \bigg)^{-1} C_1\N\]\N(if the limit exists) is called an infinite matrix continued fraction (MCF). The article considers the case when all \(C_k=I (k=1,2,\dots)\). MCF is written as\N\[\NA_0+\cfrac{B_1}{A_1+\cfrac{B_2}{A_2+\ddots}},\N\]\Nwhere \(B/A=BA^{-1}\). The problem of simultaneous block-diagonalization \(A_k,B_k\) is studied in the paper. It is well known that the matrices \(A_1,\dots, A_s\) can be simultaneously block-diagonalised if and only if the algebra \(\mathcal{A}\) is completely reducible. The semi-simplicity of \(\mathcal{A}\) is tested. The procedure for the explicit decomposition proposed. The matrix Padé approximants for matrix-valued functions \(F(z)\) are investigated in the article. A large number of illustrative examples are included in the article.