Question

When a deep space probe launched, corrections may be necessary to place the probe on a precisely calculated trajectory. Radio elementary provides a stream of vectors, \({{\bf{x}}_{\bf{1}}},....,{{\bf{x}}_k}\), giving information at different times about how the probe’s position compares with its planned trajectory. Let \({X_k}\) be the matrix \(\left[ {{x_{\bf{1}}}.....{x_k}} \right]\). The matrix \({G_k} = {X_k}X_k^T\) is computed as the radar data are analyzed. When \({x_{k + {\bf{1}}}}\) arrives, a new \({G_{k + {\bf{1}}}}\) must be computed. Since the data vector arrive at high speed, the computational burden could be serve. But partitioned matrix multiplication helps tremendously. Compute the column-row expansions of \({G_k}\) and \({G_{k + {\bf{1}}}}\) and describe what must be computed in order to update \({G_k}\) to \({G_{k + {\bf{1}}}}\).

Short answer

To achieve \({G_{k + 1}}\) from \({G_k}\), add the expression \[{\rm{Co}}{{\rm{l}}_{k + 1}}\left( {{X_{k + 1}}} \right){\rm{ro}}{{\rm{w}}_{k + 1}}\left( {X_{k + 1}^T} \right)\] to \({G_k}\).

Step-by-step solution

Solve \({G_k}\) using column-row expansion

Column-row expansion of \({G_k}\) is:

\(\begin{array}{c}{G_k} = {X_k}X_k^T\\ = {\rm{co}}{{\rm{l}}_1}\left( {{X_k}} \right){\rm{ro}}{{\rm{w}}_1}\left( {X_k^T} \right) + .... + {\rm{co}}{{\rm{l}}_k}\left( {{X_k}} \right){\rm{ro}}{{\rm{w}}_k}\left( {X_k^T} \right)\end{array}\)

Solve \({G_{k + {\bf{1}}}}\) using column-row expansion

Column-rowexpansion of \({G_{k + 1}}\) is:

\[\begin{array}{c}{G_{k + 1}} = {X_{k + 1}}X_{k + 1}^T\\ = {\rm{Co}}{{\rm{l}}_1}\left( {{X_{k + 1}}} \right){\rm{ro}}{{\rm{w}}_1}\left( {X_{k + 1}^T} \right) + .... + {\rm{Co}}{{\rm{l}}_k}\left( {{X_{k + 1}}} \right){\rm{ro}}{{\rm{w}}_k}\left( {X_{k + 1}^T} \right) + {\rm{Co}}{{\rm{l}}_{k + 1}}\left( {{X_{k + 1}}} \right){\rm{ro}}{{\rm{w}}_{k + 1}}\left( {X_{k + 1}^T} \right)\\ = {G_k} + {\rm{Co}}{{\rm{l}}_{k + 1}}\left( {{X_{k + 1}}} \right){\rm{ro}}{{\rm{w}}_{k + 1}}\left( {X_{k + 1}^T} \right)\end{array}\]

The first \(k\) columns of \({X_{k + 1}}\) are identical to the first \(k\) columns of \({X_k}\). So, to achieve \({G_{k + 1}}\) from \({G_k}\), add the expression \[{\rm{Co}}{{\rm{l}}_{k + 1}}\left( {{X_{k + 1}}} \right){\rm{ro}}{{\rm{w}}_{k + 1}}\left( {X_{k + 1}^T} \right)\] to \({G_k}\).