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

19. \(A = \left( {\begin{array}{*{20}{c}}{.9}&1&0\\0&{ - .9}&0\\0&0&{.5}\end{array}} \right),B = \left( {\begin{array}{*{20}{c}}0\\1\\1\end{array}} \right)\).

Short Answer

Expert verified

The matrix pairs \(\left( {A,B} \right)\) are controllable.

Step by step solution

01

Define the rank of a matrix

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

02

Determine the rank of the matrix

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

Write the augmented matrix as shown below:

\(\left( {\begin{array}{*{20}{c}}B&{AB}&{{A^2}B}\end{array}} \right) = \left( {\begin{array}{*{20}{c}}0&1&0\\1&{ - .9}&{.81}\\1&{.5}&{.25}\end{array}} \right)\)

Perform an elementary row operation to produce the row-reduced echelon form of the matrix.

Interchange rows 1 and 2.

\( \sim \left( {\begin{array}{*{20}{c}}1&{ - .9}&{0.81}\\0&1&0\\1&{.5}&{.25}\end{array}} \right)\)

At row 3, subtract row 1 from row 3.

\( \sim \left( {\begin{array}{*{20}{c}}1&{ - .9}&{0.81}\\0&1&0\\0&{1.4}&{ - 0.56}\end{array}} \right)\)

At row 1, multiply row 2 by 0.9 and add it to row 1. At row 3, multiply row 2 by 1.4 and subtract it from row 3.

\( \sim \left( {\begin{array}{*{20}{c}}1&0&{0.81}\\0&1&0\\0&0&{ - 0.56}\end{array}} \right)\)

At row 3, multiply row 3 by \( - \frac{1}{{0.56}}\).

\( \sim \left( {\begin{array}{*{20}{c}}1&0&{0.81}\\0&1&0\\0&0&1\end{array}} \right)\)

At row 1, multiply row 3 by 0.81 and subtract it from row 1.

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

The matrix has three pivot columns, so the rank of the matrix is 3.

03

Determine whether the matrix pairs are controllable

The pair \(\left( {A,B} \right)\) is said to becontrollable if 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 3.

Thus, the matrix pairs \(\left( {A,B} \right)\) are controllable.

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Most popular questions from this chapter

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

20. \(A = \left( {\begin{array}{*{20}{c}}{.8}&{ - .3}&0\\{.2}&{.5}&1\\0&0&{ - .5}\end{array}} \right),B = \left( {\begin{array}{*{20}{c}}1\\1\\0\end{array}} \right)\).

Exercises 23-26 concern a vector space V, a basis \(B = \left\{ {{{\bf{b}}_{\bf{1}}},....,{{\bf{b}}_n}\,} \right\}\) and the coordinate mapping \({\bf{x}} \mapsto {\left( {\bf{x}} \right)_B}\).

Show that the coordinate mapping is onto \({\mathbb{R}^n}\). That is, given any y in \({\mathbb{R}^n}\), with entries \({y_{\bf{1}}}\),….,\({y_n}\), produce u in V such that \({\left( {\bf{u}} \right)_B} = y\).

Let \({M_{2 \times 2}}\) be the vector space of all \(2 \times 2\) matrices, and define \(T:{M_{2 \times 2}} \to {M_{2 \times 2}}\) by \(T\left( A \right) = A + {A^T}\), where \(A = \left( {\begin{array}{*{20}{c}}a&b\\c&d\end{array}} \right)\).

  1. Show that \(T\)is a linear transformation.
  2. Let \(B\) be any element of \({M_{2 \times 2}}\) such that \({B^T} = B\). Find an \(A\) in \({M_{2 \times 2}}\) such that \(T\left( A \right) = B\).
  3. Show that the range of \(T\) is the set of \(B\) in \({M_{2 \times 2}}\) with the property that \({B^T} = B\).
  4. Describe the kernel of \(T\).

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 \({P^{ - 1}}\left( {PA} \right)\).)

Find a basis for the set of vectors in\({\mathbb{R}^{\bf{2}}}\)on the line\(y = {\bf{5}}x\).

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