Generalized eigenvector

Definition

A generalized eigenvector is a vector that arises when dealing with defective matrices, which do not have a complete set of linearly independent eigenvectors. It extends the concept of eigenvectors, providing a way to analyze matrices that have repeated eigenvalues but fewer than the expected number of independent eigenvectors. Generalized eigenvectors help form a complete basis for the vector space and are essential for diagonalizing or putting the matrix into Jordan form.

5 Must Know Facts For Your Next Test

1. Generalized eigenvectors are associated with the concept of chains, where each generalized eigenvector corresponds to a specific power of the matrix minus the eigenvalue times the identity matrix.
2. For a given eigenvalue, the number of generalized eigenvectors needed can be determined by the size of the largest Jordan block associated with that eigenvalue in Jordan form.
3. Generalized eigenvectors can be computed using the equation \\( (A - \lambda I)^k v = 0 \\) for some integer k, where \( v \) is the generalized eigenvector and \( \lambda \) is the corresponding eigenvalue.
4. They provide additional structure for solving systems of linear differential equations when the coefficient matrix has repeated eigenvalues.
5. Understanding generalized eigenvectors is crucial for applications in systems theory, control theory, and other areas where matrix behavior under linear transformations is analyzed.

Review Questions

• What is the significance of generalized eigenvectors in the context of defective matrices and how do they relate to traditional eigenvectors?
  • Generalized eigenvectors are significant because they address situations where a matrix is defective, meaning it lacks enough linearly independent eigenvectors. In such cases, generalized eigenvectors fill in the gaps, allowing for a complete basis to be formed. This connection helps in understanding how to diagonalize or put a matrix into Jordan form, even when traditional eigenvector methods fall short.
• How do you compute generalized eigenvectors for a given eigenvalue and what role does their computation play in forming Jordan blocks?
  • To compute generalized eigenvectors for a given eigenvalue \( \lambda \), you solve the equation \\( (A - \lambda I)^k v = 0 \\) for increasing values of k until you find enough vectors to span the corresponding eigenspace. The size of these solutions corresponds to the sizes of the Jordan blocks in Jordan form. This process illustrates how repeated roots and their multiplicities directly influence matrix properties.
• Analyze how generalized eigenvectors contribute to solving linear differential equations and their importance in practical applications.
  • Generalized eigenvectors are crucial when dealing with linear differential equations, especially when the system's matrix has repeated eigenvalues. They allow us to construct solutions that account for both standard behavior and any additional complexity introduced by repeated roots. This capability is important in various fields such as engineering and physics, where understanding system dynamics relies on accurate solutions to differential equations involving such matrices.

Related terms

eigenvalue:

A scalar that indicates how much an eigenvector is stretched or compressed during a linear transformation represented by a matrix.

Jordan form: A canonical form of a matrix that represents its structure in terms of its eigenvalues and generalized eigenvectors, particularly useful for defective matrices.

defective matrix: A square matrix that does not have a full set of linearly independent eigenvectors, typically occurring when an eigenvalue has algebraic multiplicity greater than its geometric multiplicity.