Question

If \(\lambda\) is an eigen value of an erthogonal matrix, then \(1 / \lambda\) is also its eigen value.

Short answer

Yes, \(1 / \lambda\) is also an eigenvalue of an orthogonal matrix.

Step-by-step solution

Understand Orthogonal Matrices

An orthogonal matrix is a square matrix \( Q \) such that \( Q^T Q = I \), where \( Q^T \) is the transpose of \( Q \) and \( I \) is the identity matrix. This implies that the columns (and rows) of \( Q \) are orthonormal vectors.

Eigenvalues of Orthogonal Matrices

For an orthogonal matrix \( Q \), its eigenvalues \( \lambda \) have a magnitude of 1. This is because multiplying \( Q \) with its transpose results in the identity matrix. Mathematically, if \( Qv = \lambda v \), then \( Q^TQv = Q^T\lambda v = \lambda Q^Tv = v \), leading to \( |\lambda| = 1 \).

Analyze the Conjugate Property

For an orthogonal matrix, the eigenvalues are complex numbers on the unit circle in the complex plane, i.e., \( \lambda = e^{i\theta} \) or real numbers like ±1. Hence, if \( \lambda eq 0 \), then \( \frac{1}{\lambda} = \lambda^* \), which is the complex conjugate of \( \lambda \). Thus, \( \frac{1}{\lambda} \) is indeed also an eigenvalue because it still lies on the unit circle.

Conclusion

Given that \( \lambda \) is an eigenvalue of an orthogonal matrix, we show that \( \frac{1}{\lambda} \) is necessarily also an eigenvalue due to the properties of orthogonal matrices and the unit circle. Therefore, every eigenvalue of an orthogonal matrix must have its reciprocal eigenvalue also on the unit circle.

Key concepts

Orthogonal Matrix Properties

Orthogonal matrices have some special properties that set them apart from other types of matrices. At the core, an orthogonal matrix is a square matrix where its transpose is also its inverse. This means if you have a matrix \( Q \), then \( Q^T Q = I \), where \( Q^T \) is the transpose of \( Q \) and \( I \) is the identity matrix.

One crucial implication of this property is that the columns (and rows) of the orthogonal matrix are orthonormal. What this means is that each column vector is orthogonal to the others and each has a magnitude of one. This is why the product of an orthogonal matrix with its transpose results in the identity matrix.

Here are a few more important properties of orthogonal matrices:

• The determinant of an orthogonal matrix is always either +1 or -1.
• Orthogonal matrices preserve the dot product, which means that vector lengths and angles are conserved under multiplication by an orthogonal matrix.
• They are often used in numerical stability and to perform reflections or rotations without affecting the scale.

Eigenvalue Magnitude

When studying orthogonal matrices, one interesting property is the magnitude of their eigenvalues. For an orthogonal matrix, all eigenvalues have a magnitude of 1. This is essential because it means that, even if the eigenvalue is complex, it lies on the circle with a radius of one in the complex plane.

How do we arrive at this conclusion? Suppose you have an eigenvalue equation for an orthogonal matrix \( Q \):
\( Qv = \lambda v \), where \( v \) is a non-zero vector. If you multiply both sides by \( Q^T \), you get:
\( Q^T Qv = Q^T \lambda v = \lambda Q^T v = v \).

These steps simplify to show \( |\lambda| = 1 \). This essentially means when an orthogonal matrix acts on a vector, it does not change the vector's scale, just its direction. So, the eigenvalues being of magnitude 1 confirms that orthogonal matrices maintain vector scale.

• All eigenvalues of orthogonal matrices lie on the unit circle.
• They can be real numbers like ±1 or complex in form.

Unit Circle in Complex Plane

The unit circle in the complex plane is a fascinating concept that comes into play while analyzing eigenvalues of orthogonal matrices. The unit circle is a circle with a radius of 1 centered at the origin of the complex plane. Any complex number lying on this circle is of the form \( e^{i\theta} \), where \( \theta \) is a real number representing an angle.

When a matrix is orthogonal, its eigenvalues fall onto this unit circle. This occurs because the magnitude of each eigenvalue is precisely 1, which matches the radius of the unit circle. For instance, if you have an eigenvalue \( \lambda = e^{i\theta} \), then its position on the unit circle is determined by the angle \( \theta \).

A key feature here is that if \( \lambda \) is an eigenvalue, then its reciprocal \( \frac{1}{\lambda} \) is also an eigenvalue. For complex eigenvalues, taking a reciprocal is akin to taking the complex conjugate, which for numbers on the unit circle still remains on the circle due to the magnitude being 1. This reflects the inherent stability and consistency of orthogonal matrices.

• The unit circle ensures all eigenvalues of orthogonal matrices maintain consistent magnitudes.
• The position of any complex eigenvalue is dictated by the angle \( \theta \).
• Reciprocal eigenvalues reinforce the circular symmetry of these values.