DuaL Spaces, Norms, and Inner Products

David F. Delchamps
Additional contact information
David F. Delchamps: Cornell University, School of Electrical Engineering

Chapter 5 in State Space and Input-Output Linear Systems, 1998, pp 73-88 from Springer

Abstract: Abstract In §1, it was observed that two matrices of the same size (m×n) could be added and multiplied by scalars to form new (m ×n) matrices. The way these operations are defined, in view of Definition 4.1, makes it clear that the family of all (m×n) matrices with entries in F forms a vector space over F. In fact, the dimension of this vector space is mn, since the mn matrices E (k, l) (each is of size (m×n)) whose (i, j) elements are given, for 1 ⩽ i ⩽ m and 1 ⩽ j ⩽ n, by $${\left[ {E\left( {k,l} \right)} \right]_{ij}} = {\delta _{ik}}{\delta _{jl}} = \left\{ {\begin{array}{*{20}{c}} {1 i = k,j = l} \\ {0 otherwise} \end{array}} \right.$$ and which are defined for 1⩽£ ⩽ m, 1⩽l ⩽n, form a basis for the vector space of all m×n matrices.

Date: 1998
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-1-4612-3816-4_6

Ordering information: This item can be ordered from
http://www.springer.com/9781461238164

DOI: 10.1007/978-1-4612-3816-4_6

Access Statistics for this chapter

More chapters in Springer Books from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().