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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

[M] Suppose memory or size restrictions prevent your matrix program from working with matrices having more than 32 rows and 32 columns, and suppose some project involves \(50 \times 50\) matrices A and B. Describe the commands or operations of your program that accomplish the following tasks.

a. Compute \(A + B\)

b. Compute \(AB\)

c. Solve \(Ax = b\) for some vector b in \({\mathbb{R}^{50}}\), assuming that \(A\) can be partitioned into a \(2 \times 2\) block matrix \(\left[ {{A_{ij}}} \right]\), with \({A_{11}}\) an invertible \(20 \times 20\) matrix, \({A_{22}}\) an invertible \(30 \times 30\) matrix, and \({A_{12}}\) a zero matrix. [Hint: Describe appropriate smaller systems to solve, without using any matrix inverse.]

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

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

Let T be a linear transformation that maps \({\mathbb{R}^n}\) onto \({\mathbb{R}^n}\). Is \({T^{ - 1}}\) also one-to-one?

In Exercises 33 and 34, Tis a linear transformation from \({\mathbb{R}^2}\) into \({\mathbb{R}^2}\). Show that T is invertible and find a formula for \({T^{ - 1}}\).

33. \(T\left( {{x_1},{x_2}} \right) = \left( { - 5{x_1} + 9{x_2},4{x_1} - 7{x_2}} \right)\)

Assume \(A - s{I_n}\) is invertible and view (8) as a system of two matrix equations. Solve the top equation for \({\bf{x}}\) and substitute into the bottom equation. The result is an equation of the form \(W\left( s \right){\bf{u}} = {\bf{y}}\), where \(W\left( s \right)\) is a matrix that depends upon \(s\). \(W\left( s \right)\) is called the transfer function of the system because it transforms the input \({\bf{u}}\) into the output \({\bf{y}}\). Find \(W\left( s \right)\) and describe how it is related to the partitioned system matrix on the left side of (8). See Exercise 15.
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