Eigenvector

Definition

An eigenvector is a non-zero vector that changes only by a scalar factor when a linear transformation is applied to it. In other words, if you multiply a matrix by an eigenvector, the result is simply the eigenvector scaled by a certain value known as the eigenvalue. This relationship is key to understanding how linear transformations act within vector spaces and helps reveal the structure of those spaces.

5 Must Know Facts For Your Next Test

1. Eigenvectors must be non-zero; otherwise, they do not provide meaningful information about the transformation.
2. If a matrix has n eigenvectors, it may have up to n distinct eigenvalues.
3. Eigenvectors corresponding to distinct eigenvalues are linearly independent.
4. The eigenspace of an eigenvector is the set of all scalar multiples of that eigenvector, often forming a subspace.
5. Finding eigenvectors involves solving the characteristic equation, which is derived from setting the determinant of (A - λI) equal to zero, where A is the matrix, λ is the eigenvalue, and I is the identity matrix.

Review Questions

• How do eigenvectors relate to linear transformations and what does their behavior tell us about these transformations?
  • Eigenvectors demonstrate how certain vectors maintain their direction under linear transformations while being scaled. This behavior indicates specific properties of the transformation represented by the matrix. When you multiply a matrix by its eigenvector, if the output is simply the original vector multiplied by a scalar (the eigenvalue), it reveals critical insights into the nature of the transformation, such as whether it stretches or compresses along specific directions.
• Describe how to find an eigenvector from a given matrix and explain the significance of the characteristic equation in this process.
  • To find an eigenvector for a given matrix, you first determine its eigenvalues by solving the characteristic equation derived from det(A - λI) = 0. Once you have the eigenvalues, substitute each one back into (A - λI)v = 0, where v is the eigenvector. The solutions to this equation give you the corresponding eigenvectors. The characteristic equation is significant because it encapsulates the relationships between the matrix's properties and its eigenvectors and eigenvalues.
• Evaluate how understanding eigenvectors and their corresponding eigenspaces can impact applications in fields such as physics or engineering.
  • Understanding eigenvectors and eigenspaces has profound implications in fields like physics and engineering because they simplify complex systems into manageable components. For instance, in quantum mechanics, state vectors are often described in terms of eigenvectors, which helps in analyzing quantum states and observables. In engineering, modal analysis utilizes eigenvectors to understand vibrational patterns in structures. Thus, mastering these concepts allows for better prediction and control over dynamic systems across various applications.

Related terms

Eigenvalue: An eigenvalue is a scalar associated with an eigenvector, indicating how much the eigenvector is stretched or compressed during the transformation.

Linear Transformation: A linear transformation is a mapping between two vector spaces that preserves the operations of vector addition and scalar multiplication.

Matrix Representation: The representation of a linear transformation as a matrix allows for the computation of transformations, including finding eigenvalues and eigenvectors.