Characteristic Polynomial

Definition

The characteristic polynomial of a square matrix is a polynomial that is derived from the determinant of the matrix subtracted by a variable times the identity matrix. This polynomial provides crucial information about the eigenvalues of the matrix, as the roots of the characteristic polynomial correspond to the eigenvalues. Additionally, it encapsulates important properties of the matrix, allowing for analysis in various mathematical contexts, including stability and system dynamics.

5 Must Know Facts For Your Next Test

1. The characteristic polynomial of an n x n matrix A can be expressed as $$p(\lambda) = \text{det}(A - \lambda I)$$, where I is the identity matrix.
2. The degree of the characteristic polynomial is always equal to the size of the matrix (n), and it will have n roots corresponding to the eigenvalues.
3. Finding the characteristic polynomial is a key step in determining whether a matrix is diagonalizable by analyzing its eigenvalues and their algebraic multiplicities.
4. The roots of the characteristic polynomial may include complex numbers, even if all elements of the original matrix are real.
5. The characteristic polynomial can be used to derive other important polynomials related to matrices, such as the minimal polynomial and the companion matrix.

Review Questions

• How does the characteristic polynomial relate to finding eigenvalues of a matrix?
  • The characteristic polynomial is directly linked to finding the eigenvalues of a matrix. By calculating the determinant of $$A - \lambda I$$ and setting it equal to zero, we obtain a polynomial equation whose roots are the eigenvalues. This process allows us to determine not only the values for which there are non-trivial solutions to the equation $$Ax = \lambda x$$, but also reveals insights into the behavior and properties of linear transformations represented by the matrix.
• Discuss how the characteristic polynomial can indicate whether a matrix is diagonalizable.
  • The characteristic polynomial plays a crucial role in assessing whether a matrix is diagonalizable. If the algebraic multiplicity (the number of times each eigenvalue appears as a root) equals its geometric multiplicity (the dimension of its eigenspace), then the matrix can be diagonalized. Analyzing the roots of the characteristic polynomial gives insight into these multiplicities, making it an essential tool in understanding if a linear transformation can be simplified into a diagonal form.
• Evaluate the implications of complex eigenvalues obtained from the characteristic polynomial in applications such as stability analysis.
  • Complex eigenvalues resulting from the characteristic polynomial can significantly impact stability analysis in dynamic systems. For instance, if a system's characteristic polynomial yields complex conjugate pairs with positive real parts, this indicates instability as solutions will grow unbounded over time. Conversely, complex pairs with negative real parts imply stability, where oscillatory behavior decays over time. Thus, understanding these roots helps engineers and scientists predict system behavior and make necessary adjustments for desired stability.

Related terms

Eigenvalues: Values that represent how much a linear transformation stretches or shrinks vectors along certain directions, found as the roots of the characteristic polynomial.

Eigenvectors: Non-zero vectors that change only in scale when a linear transformation is applied, associated with their corresponding eigenvalues.

Determinant: A scalar value that is a function of a square matrix and provides important information about the matrix, including whether it is invertible and its volume scaling factor.