Eigenvalue

Definition

An eigenvalue is a special number associated with a linear transformation represented by a matrix, indicating how much a corresponding eigenvector is stretched or compressed during that transformation. In seismology, eigenvalues play a crucial role in analyzing moment tensors, as they help identify the nature of the seismic sources and their focal mechanisms. By determining the eigenvalues, one can gain insight into the orientation and type of stress distribution involved in an earthquake.

5 Must Know Facts For Your Next Test

1. Eigenvalues are crucial for understanding the mechanics of seismic sources, as they help in determining whether an earthquake is strike-slip, normal, or thrust.
2. In the context of moment tensors, there are typically three eigenvalues, which can provide information about the relative sizes of the different stress components at the source.
3. The signs of the eigenvalues can reveal whether the deformation involved in an earthquake is compressive or extensional.
4. When plotting eigenvalues on a graph, one can visualize the stress regime and better understand how different forces interact during seismic events.
5. Eigenvalue analysis assists in interpreting focal mechanism solutions by relating them to the geometry of faulting and the properties of the earth's crust.

Review Questions

• How do eigenvalues relate to the analysis of moment tensors in understanding seismic sources?
  • Eigenvalues are fundamental in analyzing moment tensors because they represent key characteristics of the seismic source's stress state. Each eigenvalue corresponds to a specific direction and magnitude of stress acting at the fault. By examining these values, seismologists can classify the earthquake mechanism and understand how energy is released during seismic events.
• Discuss how changes in eigenvalues can affect our interpretation of focal mechanism solutions and what this means for our understanding of earthquake behavior.
  • Changes in eigenvalues can significantly impact the interpretation of focal mechanism solutions by altering the perceived stress distribution at an earthquake's source. For example, variations in magnitudes or signs of these eigenvalues may indicate shifts between compressional or extensional regimes. This leads to a better understanding of fault dynamics and assists researchers in predicting potential future seismic activity in specific areas.
• Evaluate the importance of eigenvalue analysis in improving seismic hazard assessments and earthquake preparedness strategies.
  • Eigenvalue analysis is critical for enhancing seismic hazard assessments because it provides valuable insights into the behavior of faults and potential earthquake mechanisms. By understanding the eigenvalues associated with various fault systems, scientists can predict how an area might respond to tectonic forces. This information helps in developing targeted preparedness strategies, ensuring that communities are better equipped to handle potential seismic risks based on identified stress regimes.

Related terms

Eigenvector: An eigenvector is a non-zero vector that changes by only a scalar factor when a linear transformation is applied, associated with its corresponding eigenvalue.

Moment Tensor: A moment tensor is a mathematical representation of the force system that causes an earthquake, describing the distribution of seismic moment in three dimensions.

Principal Axes: Principal axes refer to the orientations of the eigenvectors associated with the moment tensor, which indicate the directions of maximum and minimum stress during seismic events.