Eigenvalues

Definition

Eigenvalues are special numbers associated with a linear transformation represented by a matrix, indicating the factor by which the corresponding eigenvectors are stretched or compressed. They help understand the behavior of linear transformations and play a crucial role in various applications, including stability analysis and system dynamics.

5 Must Know Facts For Your Next Test

1. Eigenvalues can be real or complex numbers, depending on the nature of the linear transformation represented by the matrix.
2. To find the eigenvalues of a matrix, you typically solve the characteristic equation, which is formed by setting the determinant of the matrix minus a scalar multiple of the identity matrix to zero.
3. The number of eigenvalues of an n x n matrix is equal to n, although some eigenvalues may be repeated (known as algebraic multiplicity).
4. In the context of Maschke's theorem, which states that every finite group representation over a field of characteristic zero is completely reducible, eigenvalues help determine whether certain representations can be decomposed into simpler components.
5. Eigenvalues are essential for understanding the stability of systems, as they indicate whether small perturbations will grow or decay over time based on their magnitude and sign.

Review Questions

• How do eigenvalues relate to the concept of linear transformations and their effects on vectors?
  • Eigenvalues are closely tied to linear transformations as they signify how much the corresponding eigenvectors are stretched or compressed during such transformations. Specifically, for a given matrix representing a linear transformation, when an eigenvector is multiplied by this matrix, it results in a new vector that is simply a scaled version of the original. This scaling factor is precisely the eigenvalue associated with that eigenvector.
• Explain the process of finding eigenvalues using the characteristic polynomial and its significance in linear algebra.
  • To find eigenvalues, one constructs the characteristic polynomial from a square matrix by calculating the determinant of the matrix subtracted by lambda times the identity matrix. Setting this determinant to zero provides the roots of the polynomial, which are the eigenvalues. This process is significant because it not only gives insight into the properties of the transformation but also assists in various applications like diagonalization and solving differential equations.
• Analyze how Maschke's theorem utilizes eigenvalues to illustrate properties of representations in group theory.
  • Maschke's theorem asserts that any finite group representation over a field whose characteristic does not divide the order of the group can be decomposed into irreducible representations. The eigenvalues play a critical role here as they provide necessary information about the behavior of these representations under group actions. By examining how these eigenvalues behave—particularly their multiplicity and relationships—one can gain insights into whether certain representations can be simplified or combined, thus deepening our understanding of the structure of representations in group theory.

Related terms

Eigenvectors: Eigenvectors are non-zero vectors that change only in scale during a linear transformation, associated with their corresponding eigenvalues.

Characteristic Polynomial:

The characteristic polynomial is a polynomial obtained from a square matrix that is used to find the eigenvalues of that matrix.

Diagonalization: Diagonalization is the process of converting a matrix into a diagonal form using its eigenvalues and eigenvectors, simplifying many linear algebra operations.