The authors consider the block \(qd\) algorithm for block tridiagonal positive definite symmetric matrices, where the blocks are \(\ell \times \ell\) matrices. It is shown that the eigenvalues \(\lambda_i^{(k)}\), \(k \in N\), of the first block on the block diagonal of the decomposition obtained in the \(k\)th step of the \(qd\) algorithm constitute for all \(i=1,2, \dots, \ell\) a strictly increasing sequence, i.e.~\(\lambda_i^{(k)} < \lambda_i^{(k+1)}\), \(k \in N\). The eigenvalues of the last block constitute a strictly decreasing sequence. Furthermore, the convergence of the block \(qd\) algorithm is proved.