Question

Consider the system of ordinary differential equations \[ \frac{dX}{dt} = MX, \] where \( M \) is a \( 6 \times 6 \) skew-symmetric matrix with entries in \( \mathbb{R} \). Then, for this system, the origin is a stable critical point for

• A. any such matrix \( M \)
• B. only such matrices \( M \) whose rank is 2
• C. only such matrices \( M \) whose rank is 4
• D. only such matrices \( M \) whose rank is 6

Hint:

For systems with skew-symmetric matrices, the origin is always a stable critical point because the eigenvalues of a skew-symmetric matrix are purely imaginary.

Answer 1

The Correct Option is A

To determine the stability of the origin as a critical point for the given system of ordinary differential equations:

\[\frac{dX}{dt} = MX,\]

where \( M \) is a \( 6 \times 6 \) skew-symmetric matrix with entries in \( \mathbb{R} \), we need to consider the properties of skew-symmetric matrices and their eigenvalues.

1. **Definition and Properties of Skew-Symmetric Matrices:**

• A matrix \( M \) is called skew-symmetric if \( M^T = -M \).
• The eigenvalues of real skew-symmetric matrices are either zero or purely imaginary.
• This is due to the fact that if \( \lambda \) is an eigenvalue of a skew-symmetric matrix, then \( \bar{\lambda} = -\lambda \), which implies that non-zero eigenvalues occur in conjugate pairs of the form \( \pm bi \) where \( b \in \mathbb{R} \).

2. **Stability Analysis:**

• For the system 

\[\frac{dX}{dt} = MX\]

• , the origin is a critical point.
• The stability of the origin as a critical point in this linear system depends on the eigenvalues of the matrix \( M \).
• If all eigenvalues have non-positive real parts, the origin is stable.
• In the case of skew-symmetric matrices, eigenvalues are purely imaginary (non-zero) or zero, which implies that they have zero real parts.
• Thus, the origin is a stable critical point because the eigenvalues lie purely on the imaginary axis, indicating a neutral stability characterized by oscillatory behavior.

3. **Conclusion:**

As the eigenvalues of any skew-symmetric matrix have zero real parts, the origin is a stable critical point for any such matrix \( M \).

Therefore, the correct answer is:

any such matrix \( M \).