Characteristic Polynomial

Definition

The characteristic polynomial is a polynomial that is derived from a square matrix, which encapsulates information about the eigenvalues of that matrix. Specifically, it is obtained by calculating the determinant of the matrix subtracted by a scalar multiple of the identity matrix, typically expressed as $$p(\lambda) = \text{det}(A - \lambda I)$$. This polynomial is fundamental in finding the eigenvalues, as the roots of the characteristic polynomial correspond to those eigenvalues.

5 Must Know Facts For Your Next Test

1. The degree of the characteristic polynomial for an n x n matrix is n, which means it can have up to n roots corresponding to the eigenvalues.
2. The coefficients of the characteristic polynomial provide valuable insights into the properties of the eigenvalues, including their sum and product.
3. The characteristic polynomial can be used to determine if eigenvalues are real or complex; if there are complex roots, they will appear in conjugate pairs.
4. To find the eigenvalues of a matrix, one must set the characteristic polynomial equal to zero and solve for its roots.
5. In practical applications, understanding the characteristic polynomial aids in analyzing stability and dynamic behavior in systems described by matrices.

Review Questions

• How do you derive the characteristic polynomial from a given square matrix, and what does it tell you about the matrix?
  • To derive the characteristic polynomial from a square matrix A, you compute the determinant of A subtracted by lambda times the identity matrix I, expressed as $$p(\lambda) = \text{det}(A - \lambda I)$$. This polynomial helps identify the eigenvalues of the matrix, which indicate how certain vectors will be transformed by A. The roots of this polynomial provide critical information about the nature of those transformations.
• What role do the coefficients of the characteristic polynomial play in relation to its eigenvalues?
  • The coefficients of the characteristic polynomial contain important information about its eigenvalues. For example, by Vieta's formulas, the sum of the eigenvalues corresponds to the negative coefficient of the second highest degree term, while the product of all eigenvalues equals the constant term. This relationship allows one to infer properties about the eigenvalues even without directly calculating them.
• Evaluate how understanding the characteristic polynomial can influence stability analysis in dynamical systems.
  • Understanding the characteristic polynomial is crucial for stability analysis in dynamical systems because it directly relates to determining whether a system's equilibrium points are stable or unstable. By examining the eigenvalues derived from this polynomial, one can assess whether perturbations will grow or decay over time. For instance, if all eigenvalues have negative real parts, it indicates that perturbations will diminish, leading to stability. Conversely, any positive real part among the eigenvalues suggests instability, prompting further investigation into system behavior.

Related terms

Eigenvalues: Eigenvalues are scalar values that indicate how a transformation represented by a matrix stretches or compresses vectors in a particular direction.

Eigenvectors: Eigenvectors are non-zero vectors that change by only a scalar factor when a linear transformation is applied, corresponding to their respective eigenvalues.

Determinant: The determinant is a scalar value that can be computed from the elements of a square matrix, providing important properties about the matrix, such as whether it is invertible.