Table 2

\({{\boldsymbol{X}}}_{{\boldsymbol{1}}}={\boldsymbol{U}}{\boldsymbol{\Sigma }}{{\boldsymbol{V}}}^{{\boldsymbol{T}}}\)   (7)

where equation (7) shows a singular value decomposition of X₁.

\({\boldsymbol{\widetilde{{{A}}}}}={{\boldsymbol{U}}}_{{\boldsymbol{r}}}^{{{\mathbf{T}}}}{{\boldsymbol{X}}}_{{\boldsymbol{2}}}{{\boldsymbol{V}}}_{{\boldsymbol{r}}}{{\boldsymbol{\Sigma }}}_{{\boldsymbol{r}}}^{-{\boldsymbol{1}}}\) (8)

where equation (8) shows a rank-reduced representation of A projected onto the r POD modes of U.

\({\boldsymbol{\widetilde{{{A}}}}}={{\boldsymbol{Q}}}^{{{\mathbf{T}}}}{\boldsymbol{RQ}}\)   (9)

where equation (9) identifies two matrices Q and R through a Schur decomposition of \(\widetilde{{{A}}}\).

\({\boldsymbol{\varPhi }}={{\boldsymbol{X}}}_{{\boldsymbol{2}}}{{\boldsymbol{V}}}_{{\boldsymbol{r}}}\mathop{\sum }\limits_{{\boldsymbol{r}}}^{-{\boldsymbol{1}}}{\boldsymbol{Q}}\)  (10)

where equation (10) shows projected DMD modes.

\({{\boldsymbol{A}}}_{{\boldsymbol{r}}}={\boldsymbol{\varPhi }}{\boldsymbol{R}}{{\boldsymbol{\varPhi }}}^{\boldsymbol{\dagger }}.\)   (11)

where equation (11) shows a reconstruction of truncated A_r.