Patterned Linear Maps
The word “pattern” is employed in common speech to describe almost any organization of elements that exhibits some degree of non-randomness; however, in our context, the term patterned is given a more precise and narrow meaning. In this chapter, we begin by defining a patterned matrix. Our definition relies upon the identification of a base matrix or base pattern and states that the set of all patterned matrices, with respect to a given base, is simply the set of all polynomials of the base matrix. The eigenvalues and eigenvectors of patterned matrices have some notable features, which we examine. An important observation is that eigenvectors of a base matrix are also eigenvectors of all the patterned matrices generated from that base.
The definition of patterned matrices is then extended to the more general concept of patterned linear maps. We explore the relationship between the invariance of a subspace with respect to a base pattern and the invariance of the same subspace with respect to any patterned map. These invariance properties support several important results. First, we obtain that the kernel, image and eigenspaces of patterned maps are invariant with respect to the base transformation. Second, a family of patterned maps can all be decomposed by a common transformation into a set of smaller patterned maps.
The material in this chapter lays the foundation for Chapter 4, where several patterned maps are combined to represent a patterned system in state space form.
KeywordsInvariant Subspace Natural Projection Base Matrix Generalize Eigenvector Spectral Subspace
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