Chapter 2.4, Problem 17E

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 G_k = X_k $X_k^T$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.

The probe Galileo was launched October 18, 1989, and arrived near Jupiter in early December 1995.