Characteristic Polynomial

Definition

The characteristic polynomial is a polynomial derived from a square matrix that encodes important information about the matrix, particularly its eigenvalues. It is obtained by taking the determinant of the matrix subtracted by a scalar multiple of the identity matrix, and its roots correspond to the eigenvalues of the original matrix. Understanding this polynomial is crucial for solving systems of linear equations and analyzing linear transformations in mathematical economics.

5 Must Know Facts For Your Next Test

1. The characteristic polynomial of an n x n matrix is a polynomial of degree n, which means it can have up to n roots corresponding to the eigenvalues.
2. To find the characteristic polynomial, you compute the determinant of (A - λI), where A is the matrix, λ is a scalar, and I is the identity matrix.
3. The coefficients of the characteristic polynomial are related to the trace and determinant of the original matrix, linking eigenvalues to these fundamental properties.
4. Roots of the characteristic polynomial can be real or complex, providing insights into the stability and behavior of dynamic systems in economic models.
5. The Cayley-Hamilton theorem states that every square matrix satisfies its own characteristic polynomial, meaning you can use this polynomial for further analysis on the matrix.

Review Questions

• How does the characteristic polynomial relate to the eigenvalues of a matrix, and why is this connection important?
  • The characteristic polynomial is directly tied to the eigenvalues of a matrix because its roots are exactly those eigenvalues. This connection is crucial because understanding the eigenvalues allows us to analyze stability and behavior in various applications, including economic models and systems. By studying the characteristic polynomial, we can derive critical insights about how changes in input variables impact outcomes in these systems.
• Describe the process for calculating the characteristic polynomial of a given square matrix and explain its significance.
  • To calculate the characteristic polynomial of a square matrix A, you subtract λ times the identity matrix I from A to form (A - λI) and then compute its determinant. The resulting expression is a polynomial in λ. The significance lies in its ability to reveal the eigenvalues, which inform us about key properties like stability, oscillatory behavior, and equilibrium points in economic models.
• Evaluate how changes in the coefficients of a characteristic polynomial can affect its roots and what implications this has for economic systems represented by matrices.
  • Changes in the coefficients of a characteristic polynomial affect its roots because they alter the balance between terms in the polynomial equation. For instance, if a coefficient related to an eigenvalue becomes large or small, it can shift or create new roots that represent different eigenvalues. This shift has significant implications for economic systems; for example, it may indicate changes in market equilibrium or stability under different conditions, ultimately influencing decision-making processes based on these models.

Related terms

Eigenvalues: Scalar values that indicate how much a corresponding eigenvector is stretched or compressed during a linear transformation represented by a matrix.

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

Determinant: A scalar value that provides important properties of a square matrix, such as whether it is invertible and how it scales volumes in linear transformations.