In the IMSRG we are transforming a matrix via repeated applications of unitary transformation to diagonal form, that is we have at \( s=0 \) a dense matrix $$ \mathbf{A}= \begin{bmatrix} a_{11}(s=0) & a_{12}(s=0) & a_{13}(s=0) & \dots & \dots &\dots & a_{1n}(s=0) \\ a_{21}(s=0) & a_{22}(s=0)& a_{23}(s=0) & \dots & \dots &\dots &\dots \\ a_{31}(s=0) & a_{32}(s=0) & a_{33}(s=0) & a_{34}(s=0) &\dots &\dots & \dots \\ \dots & \dots & \dots & \dots &\dots &\dots & \dots\\ a_{(n-1)1}(s=0) & \dots & \dots & \dots &\dots a_{(n-1)(n-2)}(s=0) &a_{(n-1)(n-1)}(s=0) & a_{(n-1)n}(s=0)\\ a_{n1}(s=0) & \dots & \dots & \dots &\dots &a_{n(n-1)1}(s=0) & a_{nn}(s=0)\end{bmatrix}, $$
and for the next \( s \) value we may have $$ \mathbf{\tilde{A}}= \begin{bmatrix} a_{11}(s=1) & a_{12}(s=1) & 0 & 0 & \dots &0 & 0 \\ a_{21}(s=1) & a_{22}(s=1)& a_{23}(s=1) & 0 & \dots &0 &0 \\ 0 & a_{32}(s=1) & a_{33}(s=1) & a_{34}(s=1) &0 &\dots & 0\\ \dots & \dots & \dots & \dots &\dots &\dots & \dots\\ 0 & \dots & \dots & \dots &\dots a_{(n-1)(n-2)}(s=1) &a_{(n-1)(n-1)}(s=1) & a_{(n-1)n}(s=1)\\ 0 & \dots & \dots & \dots &\dots &a_{n(n-1)1}(s=1) & a_{nn}(s=1)\end{bmatrix}, $$
and finally we have $$ \mathbf{D}= \begin{bmatrix} a_{11}(s=\infty) & 0 & 0 & 0 & \dots &0 & 0 \\ 0 & a_{22}(s=\infty)& 0 & 0 & \dots &0 &0 \\ 0 & 0 & a_{33}(s=\infty) & 0 &0 &\dots & 0\\ \dots & \dots & \dots & \dots &\dots &\dots & \dots\\ 0 & \dots & \dots & \dots &\dots &a_{(n-1)(n-1)}(s=\infty) & \\ 0 & \dots & \dots & \dots &\dots &0 & a_{nn}(s=\infty)\end{bmatrix}. $$