Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

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Short Answer

a. Find all n × n matrices that are both orthogonal and upper triangular, with positive diagonal entries.
b. Show that the QR factorization of an invertible n × n matrix is unique. Hint: If, $A = Q_{1} R_{1} = Q_{2} R_{2}$ then the matrix $A = {Q_{2}}^{- 1} Q_{1} = Q_{2} {R_{1}}^{- 1}$ is both orthogonal and upper triangular, with positive diagonal entries.

n × n matrices that are both orthogonal and upper triangular, with positive diagonal entries are

A = $I_{n}$

$Q_{1} = Q_{2}$ and localid="1659498166202" $R_{1} = R_{2}$

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Consider for part (a).

If A is an orthogonal and upper triangular matrix, then $A^{- 1}$ is lower and upper triangular because $A^{- 1} = A^{T}$ and it is an inverse of an upper triangular matrix. Thus,

$A^{T} = A^{- 1}$

= A

Because A is orthogonal and diagonal with positive entries must be. So, $A = I_{n}$ .

Step 2: Consider for part (b).

Consider the theorem below.

Products and inverses of orthogonal matrices.

a. The product AB of two orthogonal n × n matrices A and B is orthogonal.

b. The inverse $A^{- 1}$ of an orthogonal n × n matrix A is orthogonal.

Thus, according to the above theorem ${Q_{2}}^{- 1} Q_{1}$ is orthogonal and $R_{2} {R_{1}}^{- 1}$ is upper triangular with positive diagonal entries.

From the part (a) it has

${Q_{2}}^{- 1} Q_{2} = R_{1} = R_{2}$

$= I_{m}$

So that, $Q_{1} = Q_{2} and R_{1} = R_{2}$ as claimed.

Hence,

n × n matrices that are both orthogonal and upper triangular, with positive diagonal entries will be

$A = I_{n}$

$Q_{1} = Q_{2} and R_{1} = R_{2}$ and both are orthogonal and upper triangular, with positive diagonal entries.

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Linear Algebra With Applications

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Step by Step Solution

Step 1: Consider for part (a).

Step 2: Consider for part (b).

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Linear Algebra With Applications

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Step by Step Solution

Step 1: Consider for part (a).

Step 2: Consider for part (b).

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