Q23E If an聽n脳n 聽matrix A is both s... [FREE SOLUTION]

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Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 8: Q23E (page 392)

If an $n 脳 n$ matrix A is both symmetric and orthogonal, what can you say about the eigenvalues of A? What about the eigenspaces? Interpret the linear transformation $T (\overset{鈫�}{x}) = A \overset{鈫�}{x}$ geometrically in the cases n = 2 and n = 3

Short Answer

Expert verified

$E_{1}$ And $E_{1}$ are orthogonal complements.

Step by step solution

The Orthogonal Matrix

An orthogonal matrix, also known as an orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors in linear algebra.
Where $Q^{T}$ is the transpose ofQand I is the identity matrix is one approach to describe this.

Determine the value of k

By definition we know that

A Is symmetric $鈬� A = A^{T}$ and A is orthogonal $鈬� A A^{T} = I$

$A A^{T} = I A A = I A^{2} = I$

Let $位$ be an eigenvalue of A, where there exists a nonzero $\overset{鈫�}{x}$ such that $A^{2} \overset{鈫�}{x} = 位^{2} \overset{鈫�}{x}$

$A^{2} \overset{鈫�}{x} = I \overset{鈫�}{x} 位^{2} \overset{鈫�}{x} = \overset{鈫�}{x} (位^{2} - 1) = 0 位 = 卤 1$

Which implies that the only possible eigenvalues for A are $卤 1$ .
Since A is symmetric, $E_{1}$ and role="math" localid="1660723209698" $E_{- 1}$ are orthogonal complements.

When n = 2: If all the eigenvalues are -1 then T represents a reflection about the origin.

If all eigenvalues are +1 then we have an identity matrix and the space remains unchanged.

Now, suppose one eigenvalue is -1 and the other is +1 then represents a reflection about a line passing through the origin (that line is $E_{1}$ ).

Otherwise, all eigenvalues are +1 and T is an identity.

When n = 3 : If all the eigenvalues are -1 , then T represents a reflection about the origin.

If all eigenvalues are +1 then we have an identity matrix and the space remains unchanged.

Now, suppose one eigenvalue is -1 and the rest are +1 then T represents a reflection about a plane passing through the origin (that plane is $E_{1}$ ).

If one eigenvalue is +1 and the rest are -1 then T represents a reflection about the line passing through the origin (that line is $E_{1}$ )

Otherwise, all eigenvalues are +1 and T is an identity.

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91影视

If an $n 脳 n$ matrix A is both symmetric and orthogonal, what can you say about the eigenvalues of A? What about the eigenspaces? Interpret the linear transformation $T (\overset{鈫�}{x}) = A \overset{鈫�}{x}$ geometrically in the cases n = 2 and n = 3

Short Answer

Step by step solution

The Orthogonal Matrix

Determine the value of k

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91影视

If ann脳n matrix A is both symmetric and orthogonal, what can you say about the eigenvalues of A? What about the eigenspaces? Interpret the linear transformation T(x鈫�)=Ax鈫�geometrically in the cases n = 2 and n = 3

Short Answer

Step by step solution

The Orthogonal Matrix

Determine the value of k

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Geometry

Theoretical and Mathematical Physics

Statistics

Discrete Mathematics

Probability and Statistics

Study anywhere. Anytime. Across all devices.

If an $n 脳 n$ matrix A is both symmetric and orthogonal, what can you say about the eigenvalues of A? What about the eigenspaces? Interpret the linear transformation $T (\overset{鈫�}{x}) = A \overset{鈫�}{x}$ geometrically in the cases n = 2 and n = 3