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Chapter 2: Q2.4-16Q (page 93)

Suppose block matrix \(A\) on the left side of (7) is invertible and \({A_{{\bf{11}}}}\) is invertible. Show that the Schur component \(S\) of \({A_{{\bf{11}}}}\) is invertible. [Hint: The outside factors on the right side of (7) are always invertible. Verify this.] When \(A\) and \({A_{{\bf{11}}}}\) are invertible, (7) leads to a formula for \({A^{ - {\bf{1}}}}\), using \({S^{ - {\bf{1}}}}\) \(A_{{\bf{11}}}^{ - {\bf{1}}}\), and the other entries in \(A\).

Short Answer

Expert verified

The Schur complement \(S\) is invertible.

Step by step solution

01

Check the matrix given in (7)

As \(A\) and \({A_{11}}\) are invertible, (7)

\(\left[ {\begin{array}{*{20}{c}}I&0\\X&I\end{array}} \right]\left[ {\begin{array}{*{20}{c}}I&0\\{ - X}&I\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}I&0\\0&I\end{array}} \right]\).

And

\(\left[ {\begin{array}{*{20}{c}}I&Y\\0&I\end{array}} \right]\left[ {\begin{array}{*{20}{c}}I&{ - Y}\\0&I\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}I&0\\0&I\end{array}} \right]\)

As the matrices \(\left[ {\begin{array}{*{20}{c}}I&0\\X&I\end{array}} \right]\) and \(\left[ {\begin{array}{*{20}{c}}I&Y\\0&I\end{array}} \right]\) are square, they are invertible.

02

Simplify equation (7)

Multiply the equation \(\left[ {\begin{array}{*{20}{c}}{{A_{11}}}&{{A_{12}}}\\{{A_{21}}}&{{A_{22}}}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}I&0\\X&I\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{{A_{11}}}&0\\0&S\end{array}} \right]\left[ {\begin{array}{*{20}{c}}I&Y\\0&I\end{array}} \right]\) by \({\left[ {\begin{array}{*{20}{c}}I&0\\X&I\end{array}} \right]^{ - 1}}\) and \({\left[ {\begin{array}{*{20}{c}}I&Y\\0&I\end{array}} \right]^{ - 1}}\) on both sides.

\({\left[ {\begin{array}{*{20}{c}}I&0\\X&I\end{array}} \right]^{ - 1}}A{\left[ {\begin{array}{*{20}{c}}I&Y\\0&I\end{array}} \right]^{ - 1}} = \left[ {\begin{array}{*{20}{c}}{{A_{11}}}&0\\0&S\end{array}} \right]\)

Since the matrix \(\left[ {\begin{array}{*{20}{c}}{{A_{11}}}&0\\0&S\end{array}} \right]\) is the product of inverse matrices, the Schur complement \(S\) is invertible.

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Most popular questions from this chapter

A useful way to test new ideas in matrix algebra, or to make conjectures, is to make calculations with matrices selected at random. Checking a property for a few matrices does not prove that the property holds in general, but it makes the property more believable. Also, if the property is actually false, you may discover this when you make a few calculations.

36. Write the command(s) that will create a \(6 \times 4\) matrix with random entries. In what range of numbers do the entries lie? Tell how to create a \(3 \times 3\) matrix with random integer entries between \( - {\bf{9}}\) and 9. (Hint:If xis a random number such that 0 < x < 1, then \( - 9.5 < 19\left( {x - .5} \right) < 9.5\).

[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.]

Let \(A = \left( {\begin{aligned}{*{20}{c}}1&1&1\\1&2&3\\1&4&5\end{aligned}} \right)\), and \(D = \left( {\begin{aligned}{*{20}{c}}2&0&0\\0&3&0\\0&0&5\end{aligned}} \right)\). Compute \(AD\) and \(DA\). Explain how the columns or rows of A change when A is multiplied by D on the right or on the left. Find a \(3 \times 3\) matrix B, not the identity matrix or the zero matrix, such that \(AB = BA\).

If Ais an \(n \times n\) matrix and the transformation \({\bf{x}}| \to A{\bf{x}}\) is one-to-one, what else can you say about this transformation? Justify your answer.

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.

16. a. If A and B are \({\bf{3}} \times {\bf{3}}\) and \(B = \left( {\begin{aligned}{*{20}{c}}{{{\bf{b}}_1}}&{{{\bf{b}}_2}}&{{{\bf{b}}_3}}\end{aligned}} \right)\), then \(AB = \left( {A{{\bf{b}}_1} + A{{\bf{b}}_2} + A{{\bf{b}}_3}} \right)\).

b. The second row of ABis the second row of Amultiplied on the right by B.

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

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

e. The transpose of a sum of matrices equals the sum of their transposes.
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