Statistical Signal Processing in Engineering

2Linear Algebraic Systems

Linear algebraic models give a compact and powerful structure for signal modeling. Filtering, identification, and estimation can be modeled by a set of liner equations, possibly using matrix notation for the unknown x:

Its recurrence in many contexts makes it mandatory to gain insight into some essential concepts beyond the pure numerical solution, as these are necessary for an in‐depth understanding of the properties of linear systems as linear transformations manipulating matrix subspaces.

2.1 Problem Definition and Vector Spaces

The linear system

(2.1)

represents the linear transformation of the vector into the vector through the linear transformation described by the A matrix where we assume that (i.e., the number of equations is larger than the number of unknowns). We can rewrite the previous equation as

this shows that b is obtained as a linear combination ...