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Chapter 12: Problem 14

Show how to compute the eigenvalues of an orthogonal matrix \(A \in \mathbf{R}^{n \times n}\) by computing the Schur decompositions of \(\left(A+A^{T}\right) / 2\) and \(\left(A-A^{T}\right) / 2\).

Short Answer

Expert verified
The eigenvalues of the orthogonal matrix \( A \) have unit modulus.

Step by step solution

01

Understand Orthogonal Matrix Properties

An orthogonal matrix \( A \) is a real square matrix where \( A^T A = I \), where \( I \) is the identity matrix. Its eigenvalues have modulus 1.
02

Form the Symmetric Matrix

Compute \( B = \frac{A + A^T}{2} \). This matrix is symmetric because \( (B)^T = B \). Symmetric matrices have real eigenvalues.
03

Find the Skew-Symmetric Matrix

Compute \( C = \frac{A - A^T}{2} \). This matrix is skew-symmetric as \( (C)^T = -C \). Eigenvalues of skew-symmetric matrices are purely imaginary or zero.
04

Use Schur Decomposition on Symmetric Matrix

Decompose the symmetric matrix \( B \) using the Schur decomposition: \( B = Q_T R_T Q_T^T \), where \( Q_T \) is orthogonal and \( R_T \) is upper triangular. Since \( B \) is symmetric, \( R_T \) will be diagonal with real entries (the eigenvalues of \( B \)).
05

Use Schur Decomposition on Skew-Symmetric Matrix

Decompose the skew-symmetric matrix \( C \) using the Schur decomposition: \( C = Q_S R_S Q_S^T \) with \( Q_S \) orthogonal and \( R_S \) upper triangular. Since \( C \) is skew-symmetric, the eigenvalues of \( R_S \) will be purely imaginary or zero.
06

Combine the Results to Find Eigenvalues of A

The eigenvalues of \( A \) are derived from the eigenvalues of \( B + iC \), \( i \) being the imaginary unit. They are of the form \( \lambda = \cos(\theta) + i\sin(\theta) \), as \( B + iC \) represents a rotation, ensuring all \( \lambda \) have modulus 1.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Schur Decomposition
The Schur decomposition is a valuable tool in linear algebra for simplifying matrix operations. It allows the transformation of any square matrix into a form where it can be easily analyzed.

For any square matrix, the Schur decomposition can be represented as:
  • Matrix \( A \) is decomposed into \( A = Q T Q^T \),
  • where \( Q \) is an orthogonal matrix containing the eigenvectors, and
  • \( T \) is an upper triangular matrix containing the eigenvalues of \( A \).
The orthogonal matrix \( Q \) ensures that the transformation preserves lengths, and the matrix \( T \) provides direct access to the eigenvalues without further complex operations. This decomposition is particularly insightful for symmetric and skew-symmetric matrices, where the properties of matrix \( T \) are more specific.
Symmetric Matrices
Symmetric matrices are a special category of matrices where the matrix is equal to its own transpose. In mathematical terms, a square matrix \( B \) is symmetric if \( B = B^T \).

Some key characteristics of symmetric matrices include:
  • Real eigenvalues: The eigenvalues of a symmetric matrix are always real numbers.
  • Diagonalizability: Symmetric matrices are always diagonalizable. This means they can be represented in the form \( B = Q \, \Lambda \, Q^T \), where \( Q \) is orthogonal and \( \Lambda \) is a diagonal matrix of eigenvalues.
  • Positive Semi-Definiteness: Symmetric matrices have important implications in optimization, especially in determining types of critical points.
Symmetric matrices play an essential role in physics and engineering, often describing systems with no preferred direction.
Skew-Symmetric Matrices
Skew-symmetric matrices, in contrast to symmetric matrices, satisfy the property \( C = -C^T \). This means that each element is the negative of its transpose counterpart. Some distinctive traits of skew-symmetric matrices include:

  • Purely Imaginary Eigenvalues: Skew-symmetric matrices have eigenvalues that are purely imaginary, except for the possibility of zero.
  • Odd-Dimensional Zero Sum: In the case of skew-symmetric matrices with an odd dimension, they must have at least one zero eigenvalue.
  • Characteristic in Rotations: In mechanics and other applied fields, skew-symmetric matrices often define rotational transformations.
Like symmetric matrices, the decomposition of skew-symmetric matrices helps in simplifying complex computations, giving us insight into systems exhibiting rotational symmetries or anti-symmetries.
Orthogonal Matrices
Orthogonal matrices are special matrices where the rows and columns are orthonormal vectors. An orthogonal matrix \( A \) fulfills the condition \( A^T A = I \), where \( I \) is the identity matrix. This property ensures several interesting features:

  • Preservation of Vector Norms: Multiplying a vector by an orthogonal matrix does not change the vector's length.
  • Eigenvalues of Unit Magnitude: The eigenvalues of an orthogonal matrix have a modulus of one, typically of the form \( e^{i\theta} \).
  • Stability in Computations: In numerical algorithms, orthogonal matrices help maintain stability and prevent errors from growing.
  • Applications in 3D Rotations: Commonly used to represent rotations in three-dimensional space.
In linear algebra, orthogonal matrices are instrumental in preserving geometric properties under transformations, making them invaluable in computations involving rotations and reflections.

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Most popular questions from this chapter

(a) If \(T \in \mathbf{R}^{n \times n}\) is Toeplitz, show how to compute \(R, S \in \mathbf{R}^{n \times 2}\) so that \(Z_{1} T-T Z_{-1}=R S^{T}\). (b) Suppose \(R, S \in \mathbf{R}^{n \times r}\) and \(T \in \mathbf{R}^{n \times n}\) satisfy \(Z_{1} T-T Z_{-1}=R S^{T}\). Give an algorithm that computes \(\mathrm{u}=T(\mathrm{l})\) and \(v=T(1, \xi)^{T} .\)

Suppose \(B \in \mathbf{R}^{n \times n}\) and \(C \in \mathbf{R}^{m \times m}\) are unsymmetric and positive definite. Does it follow that \(B \otimes C\) is positive definite?

Suppose \(A \in \mathbf{R}^{n_{1}} \times \cdots \times n_{d}\) and that \(\mathcal{A}_{(k)}\) has unit rank for some \(k\). Does it follow that \(A\) is a rank-1 tensor?

How many fibers are there in the tensor \(A \in \mathbf{R}^{n_{1} x \cdots \times n_{u}}\) ? How many slices?

Suppose \(B \in \mathbf{R}^{n \times n}\) and \(C \in \mathbf{R}^{m \times m}\) are unsymmetric and positive definite. Does it follow that \(B \otimes C\) is positive definite?
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