When a deep space probe is launched, corrections may be necessary to place the probe on...

When a deep space probe is launched, corrections may be necessary to place the probe on a precisely calculated trajectory. Radio telemetry provides a stream of vectors, x₁,…, x_k giving information at different times about how the probe’s position compares with its planned trajectory. Let X_k be the matrix [x₁… x_k]. The matrix

is computed as the radar data are analyzed. When x_{k+ 1} arrives, a new G_k+1 must be computed. Since the data vectors arrive at high speed, the computational burden could be severe. But partitioned matrix multiplication helps tremendously. Compute the column–row expansions of G_k and G_k+1, and describe what must be computed in order to update G_k to form G_k+1.