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Exercises 12–17 develop properties of rank
Chapter , Problem 17E(choose chapter or problem)
Problem 17E
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).
Questions & Answers
QUESTION:
Problem 17E
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).
ANSWER:Solution 17E
Step 1 of 2
Consider the following statement:
“A submatrix of a matrix A is any matrix that results from deleting some rows and/or
columns of A. It can be shown that the matrix A has rank r if and only if A contains an
invertible submatrix and no larger square matrix is invertible”.
(a)
The objective is to demonstrate this statement by explaining why an matrix A of rank r has an submatrix or rank r.
Let A be the matrix of order with rank r.
Let the matrixconsist r pivot columns of A.
Since the matrixconsist r pivot columns, the columns of are linearly independent.
Therefore, the matrix is of order with rank r.