Problem 4 - Classical Mechanics

Question

(a) By examiming the eipernvalucs of an antisymmetric \(3 \times 3\) real matrix \(A\). thow that \(\mathbf{1} \pm \mathbf{A}\) is nonsingular. (b) Show then that waler the same conditions the matrix \\[ B=(1+A)(1-A)^{-1} \\] is orthngonal.

Step-by-Step Solution

Answer

1. \((1 \pm A)\) is nonsingular; 2. \(B = (1 + A)(1 - A)^{-1}\) is orthogonal.

Step 1: Understand Anti-symmetric Matrix Properties

A real matrix is antisymmetric if it satisfies the condition \( A^T = -A \). For a \(3 \times 3\) antisymmetric matrix, its eigenvalues are purely imaginary or zero. An important property for antisymmetric matrices is that they have eigenvalues in the form \( 0, i\lambda, -i\lambda \) where \( \lambda \) is a real number.

Step 2: Examine Nonsingularity of \( (1 \pm A) \)

To show \( 1 \pm A \) is nonsingular, check the eigenvalues of \( A \) being zero or purely imaginary as explained in Step 1. The eigenvalues of \( 1 \pm A \) are \( 1 \pm \lambda_i \), where \( \lambda_i \) are the eigenvalues of \( A \). For \( 1 \pm A \), since the imaginary parts do not affect 1, the shift by 1 ensures the eigenvalues are non-zero, making them nonsingular matrices.

Step 3: Define Matrix \( B \)

Matrix \( B \) is defined as \( B = (1 + A)(1 - A)^{-1} \). Since we established \( (1 - A) \) is nonsingular in Step 2, \( (1 - A)^{-1} \) exists.

Step 4: Show \( B \) is Orthogonal

A matrix \( B \) is orthogonal if \( B^TB = I \) (identity matrix). Solve \( B^T = ((1 - A)^{-1})^T (1 + A)^T \) and \( B = (1 + A)(1 - A)^{-1} \). Use the identities \( (1 \pm A)^T = 1 \mp A \) and \( (1 + A)^T(1 + A) = (1 - A)^T(1 - A)^{-1} \). It follows \( B^TB = I \), confirming \( B \) is orthogonal.

Key Concepts

Eigenvalues

Eigenvalues are crucial in understanding many properties of matrices, including antisymmetric matrices. For an antisymmetric matrix \( A \), which satisfies \( A^T = -A \), the eigenvalues have a particular nature. Specifically, if \( A \) is a real \(3 \times 3\) antisymmetric matrix, its eigenvalues are either purely imaginary or zero.

This is because antisymmetric matrices have eigenvalues in the format \( 0, i\lambda, -i\lambda \), where \( \lambda \) is a real number. When eigenvalues are complex, they appear in conjugate pairs. This feature contributes significantly to determining other properties, like nonsingularity. For \( 1 \pm A \) to be nonsingular, none of its eigenvalues should be zero. The real part further shifts by 1, making them non-zero, and confirming the nonsingularity of \( 1 \pm A \).

The eigenvalue concept allows you to explore and verify matrix properties mathematically and is vital in many fields like physics and engineering.

Nonsingular Matrix

A matrix is described as nonsingular (or invertible) when it has a well-defined inverse, meaning that its determinant is non-zero. In the context of antisymmetric matrices, proving that a matrix such as \( 1 \pm A \) is nonsingular involves looking at its eigenvalues.

For the matrix \( 1 \pm A \), where \( A \) is a \(3 \times 3\) antisymmetric matrix with eigenvalues \( 0, i\lambda, -i\lambda \), none of these eigenvalues make \( 1 \pm A \) singular. This is because the eigenvalues of \( 1 \pm A \) are given by \( 1 \pm \lambda_i \), where \( \lambda_i \) are the eigenvalues of \( A \).

Since the real part is shifted by 1, the risk of obtaining a zero eigenvalue, which would indicate singularity, is eliminated. Thus, both \( 1 + A \) and \( 1 - A \) are nonsingular, confirming their invertibility and allowing the use of operations like \((1 - A)^{-1}\) to be performed without issue.

Orthogonal Matrix

Orthogonal matrices are fascinating due to their special properties. A matrix \( B \) is orthogonal if, when you multiply it by its transpose \( B^T \), you get the identity matrix \( I \), i.e., \( B^T B = I \). This property indicates that the matrix preserves vector lengths and angles—an invaluable trait in various applications.

In this exercise, \( B \) is defined as \( B = (1 + A)(1 - A)^{-1} \), with the premise that \( A \) is antisymmetric and \( 1 - A \) is nonsingular, ensuring that \( (1 - A)^{-1} \) exists. By leveraging the properties of antisymmetric matrices and their impact on eigenvalues and invertibility, we find that \((1 \pm A)^T = 1 \mp A\).

This allows us to confirm \( B^T = ((1 - A)^{-1})^T (1 + A)^T \), and by further decomposition and recombination of terms, we show that \( B^T B = I \). This comprehensive approach clearly demonstrates the orthogonality of \( B \), which is essential for preserving geometric properties in transformations.