Q8E If A is an n脳n聽matrix such tha... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 5: Q8E (page 263)

If A is an $n 脳 n$ matrix such that role="math" localid="1659514225617" ${AA}^{T} = I_{n}$ , then A must be an orthogonal matrix.

Short Answer

Expert verified

The given statement is true.

Step by step solution

Recollect the properties of the orthogonal a matrix.

According to the properties of a orthogonal matrix, a $n 脳 n$ matrix Ais an orthogonal matrix, if $||A \overset{鈬赌}{x}|| = ||\overset{鈬赌}{x}|| for all \overset{鈬赌}{x} 鈭� R^{n}$ .

The relation between dot and matrix product is defined as $\overset{鈬赌}{v} . \overset{鈬赌}{w} = {\overset{鈬赌}{v}}^{T} \overset{鈬赌}{w} .$

Check whether the given statement is a true or false.

The given statement is if A is an $n 脳 n$ matrix such that $A A^{T} = I_{n}$ , then A must be orthogonal matrix.

To prove A is orthogonal, prove that $||A \overset{鈬赌}{x}|| = ||\overset{鈬赌}{x}||$ .

${||A \overset{鈬赌}{x}||}^{2} = (A \overset{鈬赌}{x}) . (A \overset{鈬赌}{x}) = {(A \overset{鈬赌}{x})}^{T} (A \overset{鈬赌}{x}) = {(\overset{鈬赌}{x})}^{T} A^{T} (A \overset{鈬赌}{x}) = {(\overset{鈬赌}{x})}^{T} A^{T} A \overset{鈬赌}{x}$

Solve further as,

${||A \overset{鈬赌}{x}||}^{2} = {(\overset{鈬赌}{x})}^{T} (A^{T} A) \overset{鈬赌}{x} = {(\overset{鈬赌}{x})}^{T} (I_{n}) \overset{鈬赌}{x} = {(\overset{鈬赌}{x})}^{T} \overset{鈬赌}{x} = \overset{鈬赌}{x .} \overset{鈬赌}{x}$

Solve further as,

${||A \overset{鈬赌}{x}||}^{2} = {||\overset{鈬赌}{x}||}^{2} ||A \overset{鈬赌}{x}|| = ||\overset{鈬赌}{x}||$

So, $||A \overset{鈬赌}{x}|| = ||\overset{鈬赌}{x}||$

This means, A is orthogonal.

Then the given statement is true.

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91影视

If A is an $n 脳 n$ matrix such that role="math" localid="1659514225617" ${AA}^{T} = I_{n}$ , then A must be an orthogonal matrix.

Short Answer

Step by step solution

Recollect the properties of the orthogonal a matrix.

Check whether the given statement is a true or false.

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91影视

If A is an n脳nmatrix such that role="math" localid="1659514225617" AAT=In, then A must be an orthogonal matrix.

Short Answer

Step by step solution

Recollect the properties of the orthogonal a matrix.

Check whether the given statement is a true or false.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Discrete Mathematics

Pure Maths

Decision Maths

Statistics

Mechanics Maths

Study anywhere. Anytime. Across all devices.

If A is an $n 脳 n$ matrix such that role="math" localid="1659514225617" ${AA}^{T} = I_{n}$ , then A must be an orthogonal matrix.