cises 12–17 develop properties of rank that are sometimes

Chapter , Problem 17E

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Exercises 12–17 develop properties of rank that are sometimes needed in applications. Assume the matrix A is m × n.A submatrix of a matrix A is any matrix that results from deleting some (or no) rows and/or columns of A. It can be shown that A has rank r if and only if A contains an invertible r × r submatrix and no larger square submatrix is invertible. Demonstrate part of this statement by explaining (a) why an m × n matrix A of rank r has an m × r submatrix A1 of rank r, and (b) why A1 has an invertible r × r submatrix A2.The concept of rank plays an important role in the design of engineering control systems, such as the space shuttle system mentioned in this chapter’s introductory example. A state-space model of a control system includes a difference equation of the form where is a sequence of “state vectors” in that describe the state of the system at discrete times, and is a control, or input, sequence. The pair (A, B) is said to be controllable if The matrix that appears in (2) is called the controllability matrix for the system. If (A, B) is controllable, then the system can be controlled, or driven from the state 0 to any specified state v (in ) in at most n steps, simply by choosing an appropriate control sequence in . This fact is illustrated in Exercise 18 for n = 4 and m = 2. For a further discussion of controllability, see this text’s web site (Case Study for Chapter 4).