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Q50E

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Found in: Page 234

### Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974

# 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, ${\mathbit{A}}{\mathbf{=}}{{\mathbit{Q}}}_{{\mathbf{1}}}{{\mathbit{R}}}_{{\mathbf{1}}}{\mathbf{=}}{{\mathbit{Q}}}_{{\mathbf{2}}}{{\mathbit{R}}}_{{\mathbf{2}}}$ then the matrix ${\mathbit{A}}{\mathbf{=}}{{{\mathbit{Q}}}_{{\mathbf{2}}}}^{\mathbf{-}\mathbf{1}}{{\mathbit{Q}}}_{{\mathbf{1}}}{\mathbf{=}}{{\mathbit{Q}}}_{{\mathbf{2}}}{{{\mathbit{R}}}_{{\mathbf{1}}}}^{\mathbf{-}\mathbf{1}}$ is both orthogonal and upper triangular, with positive diagonal entries.

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

A = ${I}_{n}$

1. ${Q}_{1}={Q}_{2}$ and localid="1659498166202" ${R}_{1}={R}_{2}$
See the step by step solution

## 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 ${{\mathbit{A}}}^{\mathbf{-}\mathbf{1}}$ 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, ${Q}_{1}={Q}_{2}\mathrm{and}{R}_{1}={R}_{2}$ as claimed.

Hence,

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

$A={I}_{n}$

1. ${Q}_{1}={Q}_{2}\mathrm{and}{R}_{1}={R}_{2}$ and both are orthogonal and upper triangular, with positive diagonal entries.