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Chapter 2: Q2.4-26Q (page 93)

[M] For block operations, it may be necessary to access or enter submatrices of a large matrix. Describe the functions or commands of your matrix program that accomplish the following tasks. Suppose A is a \(20 \times 30\) matrix.

  1. Display the submatrix of Afrom rows 15 to 20 and columns 5 to 10.
  2. Insert a \(5 \times 10\) matrix B into A, beginning at row 10 and column 20.
  3. Create a \(50 \times 50\) matrix of the form \(B = \left[ {\begin{array}{*{20}{c}}A&0\\0&{{A^T}}\end{array}} \right]\).

[Note: It may not be necessary to specify the zero blocks in B.]

Short Answer

Expert verified
  1. \( > > {\mathop{\rm A}\nolimits} \left( {15:20,\,\,5:10} \right)\)
  2. \( > > {\mathop{\rm A}\nolimits} \left( {10:14,20:29} \right) = {\mathop{\rm B}\nolimits} ;\)
  3. \[ > > {\mathop{\rm B}\nolimits} = \left[ {{\mathop{\rm A}\nolimits} \,\,{\mathop{\rm zeros}\nolimits} \left( {30,20} \right);\,\,{\mathop{\rm zeros}\nolimits} \left( {20,30} \right)A'} \right];\]

Step by step solution

01

Display the submatrix of A from rows 15 to 20 and columns 5 to 10

a)

Use the MATLAB code to display the submatrix of A from rows 15 to 20, as shown below:

\( > > {\mathop{\rm A}\nolimits} \left( {15:20,\,\,5:10} \right)\)

02

Insert a \(5 \times 10\) matrix B into A

b)

Use the MATLAB code to insert a\(5 \times 10\)matrix B into A as shown below:

\( > > {\mathop{\rm A}\nolimits} \left( {10:14,20:29} \right) = {\mathop{\rm B}\nolimits} ;\)

03

Create a \(50 \times 10\) matrix

c)

Use the MATLAB code to build matrix \(B\)out of four blocks, as shown below:

\[ > > {\mathop{\rm B}\nolimits} = \left[ {{\mathop{\rm A}\nolimits} \,\,{\mathop{\rm zeros}\nolimits} \left( {30,20} \right);\,\,{\mathop{\rm zeros}\nolimits} \left( {20,30} \right)A'} \right];\]

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Most popular questions from this chapter

Suppose Ais a \(3 \times n\) matrix whose columns span \({\mathbb{R}^3}\). Explain how to construct an \(n \times 3\) matrix Dsuch that \(AD = {I_3}\).

Suppose the transfer function W(s) in Exercise 19 is invertible for some s. It can be showed that the inverse transfer function \(W{\left( s \right)^{ - {\bf{1}}}}\), which transforms outputs into inputs, is the Schur complement of \(A - BC - s{I_n}\) for the matrix below. Find the Sachur complement. See Exercise 15.

\(\left[ {\begin{array}{*{20}{c}}{A - BC - s{I_n}}&B\\{ - C}&{{I_m}}\end{array}} \right]\)

Exercises 15 and 16 concern arbitrary matrices A, B, and Cfor which the indicated sums and products are defined. Mark each statement True or False. Justify each answer.

15. a. If A and B are \({\bf{2}} \times {\bf{2}}\) with columns \({{\bf{a}}_1},{{\bf{a}}_2}\) and \({{\bf{b}}_1},{{\bf{b}}_2}\) respectively, then \(AB = \left( {\begin{aligned}{*{20}{c}}{{{\bf{a}}_1}{{\bf{b}}_1}}&{{{\bf{a}}_2}{{\bf{b}}_2}}\end{aligned}} \right)\).

b. Each column of ABis a linear combination of the columns of Busing weights from the corresponding column of A.

c. \(AB + AC = A\left( {B + C} \right)\)

d. \({A^T} + {B^T} = {\left( {A + B} \right)^T}\)

e. The transpose of a product of matrices equals the product of their transposes in the same order.

In Exercises 1–9, assume that the matrices are partitioned conformably for block multiplication. Compute the products shown in Exercises 1–4.

1. \(\left[ {\begin{array}{*{20}{c}}I&{\bf{0}}\\E&I\end{array}} \right]\left[ {\begin{array}{*{20}{c}}A&B\\C&D\end{array}} \right]\)

In Exercises 1–9, assume that the matrices are partitioned conformably for block multiplication. In Exercises 5–8, find formulas for X, Y, and Zin terms of A, B, and C, and justify your calculations. In some cases, you may need to make assumptions about the size of a matrix in order to produce a formula. [Hint:Compute the product on the left, and set it equal to the right side.]

5. \[\left[ {\begin{array}{*{20}{c}}A&B\\C&{\bf{0}}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}I&{\bf{0}}\\X&Y\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{\bf{0}}&I\\Z&{\bf{0}}\end{array}} \right]\]
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