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Chapter 4: Q 22SE (page 191)

Question: Determine if the matrix pairs in Exercises 19-22 are controllable.

22. (M) \(A = \left( {\begin{array}{*{20}{c}}0&1&0&0\\0&0&1&0\\0&0&0&1\\{ - 1}&{ - 13}&{ - 12.2}&{ - 1.5}\end{array}} \right),B = \left( {\begin{array}{*{20}{c}}1\\0\\0\\{ - 1}\end{array}} \right)\).

Short Answer

Expert verified

The matrix pair \(\left( {A,B} \right)\) is controllable.

Step by step solution

01

Define the rank of a matrix

Therank of matrix \(A\), denoted by rank\(A\), is thedimension of the column spaceof \(A\).

02

Write the augmented matrix

Calculate the rank of the matrix \(\left( {\begin{array}{*{20}{c}}B&{AB}&{{A^2}B}&{{A^3}B}\end{array}} \right)\) to determine whether the matrix pair \(\left( {A,B} \right)\) is controllable.

Write the augmented matrix as shown below:

\(\left( {\begin{array}{*{20}{c}}B&{AB}&{{A^2}B}&{{A^3}B}\end{array}} \right) = \left( {\begin{array}{*{20}{c}}1&0&0&{ - 1}\\0&0&{ - 1}&{.5}\\0&{ - 1}&{.5}&{11.45}\\{ - 1}&{.5}&{11.45}&{ - 10.275}\end{array}} \right)\)

03

Convert the matrix into row-reduced echelon form

Consider the matrix \(A = \left( {\begin{array}{*{20}{c}}1&0&0&{ - 1}\\0&0&{ - 1}&{.5}\\0&{ - 1}&{.5}&{11.45}\\{ - 1}&{.5}&{11.45}&{ - 10.275}\end{array}} \right)\).

Use the code in MATLAB to obtain the row-reduced echelon form of the matrix.

\(\begin{array}{l} > > {\mathop{\rm A}\nolimits} = \left( {1\,\,\,\,0\,\,\,\,0\,\,\, - 1;\,0\,\,\,0\,\,\, - 1\,\,\,.5;\,0\,\,\, - 1\,\,\,.5\,\,\,11.45;\,\, - 1\,\,\,.5\,\,\,11.45\,\,\, - 10.275} \right)\\ > > {\mathop{\rm U}\nolimits} = {\mathop{\rm rref}\nolimits} \left( {\mathop{\rm A}\nolimits} \right)\end{array}\)

\(A = \left( {\begin{array}{*{20}{c}}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{array}} \right)\)

The matrix has four pivot columns, so the rank of the matrix is 4.

04

Determine whether the matrix pairs are controllable

The pair \(\left( {A,B} \right)\) is said to becontrollableif rank\(\left( {\begin{array}{*{20}{c}}B&{AB}&{{A^2}B}& \cdots &{{A^{n - 1}}B}\end{array}} \right) = n\).

The rank of the matrix is 4. Therefore, the matrix pair \(\left( {A,B} \right)\) is controllable.

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Question: Exercises 12-17 develop properties of rank that are sometimes needed in applications. Assume the matrix \(A\) is \(m \times n\).

13. Show that if \(P\) is an invertible \(m \times m\) matrix, then rank\(PA\)=rank\(A\).(Hint: Apply Exercise12 to \(PA\) and \({P^{ - 1}}\left( {PA} \right)\).)

[M] Let \(A = \left[ {\begin{array}{*{20}{c}}7&{ - 9}&{ - 4}&5&3&{ - 3}&{ - 7}\\{ - 4}&6&7&{ - 2}&{ - 6}&{ - 5}&5\\5&{ - 7}&{ - 6}&5&{ - 6}&2&8\\{ - 3}&5&8&{ - 1}&{ - 7}&{ - 4}&8\\6&{ - 8}&{ - 5}&4&4&9&3\end{array}} \right]\).

  1. Construct matrices \(C\) and \(N\) whose columns are bases for \({\mathop{\rm Col}\nolimits} A\) and \({\mathop{\rm Nul}\nolimits} A\), respectively, and construct a matrix \(R\) whose rows form a basis for Row\(A\).
  2. Construct a matrix \(M\) whose columns form a basis for \({\mathop{\rm Nul}\nolimits} {A^T}\), form the matrices \(S = \left[ {\begin{array}{*{20}{c}}{{R^T}}&N\end{array}} \right]\) and \(T = \left[ {\begin{array}{*{20}{c}}C&M\end{array}} \right]\), and explain why \(S\) and \(T\) should be square. Verify that both \(S\) and \(T\) are invertible.
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