The authors extent properties that are well-known for Krylov subspace methods for solving nonsingular sparse linear systems \(A x = b\), \(A \in \mathbb{C}^{N \times N}, \enspace x, b \in \mathbb{C}^N\), to so-called block Krylov subspace methods applied to nonsingular sparse linear systems with \(s\) right-hand sides \(A X = B\), \(A \in \mathbb{C}^{N \times N}, \enspace X, B \in \mathbb{C}^{N \times s}\), with the following main results.
Using a Krylov subspace method in exact computer arithmetic generally the true solution of linear systems is found after \(\nu\) iterations, where \(\nu\) is the dimension of the largest Krylov subspace generated by \(A\) from \(r_0 = b - A x_0\), the initial residuum. Analogue to this grade \(\nu\) of \(r_0\) with respect to \(A\) (independent of the method) a block grade \(\nu_B\) of \(R_0\), the matrix of the \(s\) corresponding initial residual columns of \(A X = B\), with respect to \(A\) is defined, and it is proved that the theorem about the true solution in exact arithmetic also holds in the block Krylov subspace method.
Compared with the Krylov subspace method alternative approaches for the proofs relating to the block grade have to be used. The possibility of linear dependence in block Krylov methods is treated with special care.