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Math / Linear Algebra and Its Applications 5 / Chapter 2.4 / Problem 14E

Linear Algebra and Its Applications | 5th Edition | ISBN: 9780321982384 | Authors: David C. Lay; Steven R. Lay; Judi J. McDonald

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Show that the block upper triangular matrix A in Example 5

Chapter 2, Problem 14E

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QUESTION:

Problem 14E

Show that the block upper triangular matrix A in Example 5 is invertible if and only if both A11 and A22 are invertible. [Hint: If A11 and A22 are invertible, the formula for A–1 given in Example 5 actually works as the inverse of A.] This fact about A is an important part of several computer algorithms that estimate eigenvalues of matrices. Eigenvalues are discussed in Chapter 5.

Example 5:

A matrix of the form

is said to be block upper triangular. Assume that A11 is p × p, A22 is q × q, and A is invertible. Find a formula for A–1.

Questions & Answers

QUESTION:

Problem 14E

Show that the block upper triangular matrix A in Example 5 is invertible if and only if both A11 and A22 are invertible. [Hint: If A11 and A22 are invertible, the formula for A–1 given in Example 5 actually works as the inverse of A.] This fact about A is an important part of several computer algorithms that estimate eigenvalues of matrices. Eigenvalues are discussed in Chapter 5.

Example 5:

A matrix of the form

is said to be block upper triangular. Assume that A11 is p × p, A22 is q × q, and A is invertible. Find a formula for A–1.

ANSWER:

A matrix of the form

is said to be block upper triangular. Assume that ${A}_{11}$ is p × p, ${A}_{22}$ is q × q, and A is invertible. Then

… (1)

Solution

Step 1

In this problem we need to show that the block upper triangular matrix A is invertible if and only if both ${A}_{11}$ and ${A}_{22}$ are invertible.

Given : block upper triangular matrix,

To prove: A is invertible if and only if both ${A}_{11}$ and ${A}_{22}$ are invertible.

First let us prove: $A$ is invertible $\Rightarrow{}$ both ${A}_{11}$ and ${A}_{22}$ are invertible.

Assume $A$ is invertible.

Then by (1), we have

and exists

Hence ${A}_{11}$ and ${A}_{22}$ are invertible.

Thus $A$ is invertible $\Rightarrow{}$ both ${A}_{11}$ and ${A}_{22}$ are invertible. … (2)

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