Algebraic Multiplicity

Definition

Algebraic multiplicity refers to the number of times an eigenvalue appears as a root of the characteristic polynomial of a matrix. It indicates the multiplicity with which an eigenvalue is counted when determining the eigenvalues of a linear transformation. This concept helps in understanding the structure of the corresponding eigenspaces and the behavior of the matrix under various transformations.

5 Must Know Facts For Your Next Test

1. Algebraic multiplicity can be greater than or equal to geometric multiplicity for each eigenvalue of a matrix.
2. The sum of the algebraic multiplicities of all distinct eigenvalues equals the dimension of the vector space for square matrices.
3. A repeated eigenvalue indicates that it appears multiple times in the characteristic polynomial, which directly relates to its algebraic multiplicity.
4. Algebraic multiplicity does not provide information about the number of independent eigenvectors available for an eigenvalue, which is where geometric multiplicity comes into play.
5. The concept of algebraic multiplicity is crucial in determining if a matrix is diagonalizable; if all algebraic multiplicities equal their corresponding geometric multiplicities, then the matrix can be diagonalized.

Review Questions

• How does algebraic multiplicity relate to geometric multiplicity, and why is this relationship important?
  • Algebraic multiplicity indicates how many times an eigenvalue appears in the characteristic polynomial, while geometric multiplicity shows how many linearly independent eigenvectors correspond to that eigenvalue. This relationship is important because it helps us understand the structure of eigenspaces. For example, if the algebraic multiplicity exceeds geometric multiplicity, it suggests that not enough independent eigenvectors are present to form a complete basis for diagonalization, impacting the behavior of the matrix under linear transformations.
• Describe how to determine the algebraic multiplicity of an eigenvalue from a given characteristic polynomial.
  • To determine the algebraic multiplicity of an eigenvalue from a characteristic polynomial, you first need to factor the polynomial into its roots. The algebraic multiplicity of each distinct root corresponds to its exponent in this factored form. For instance, if the characteristic polynomial can be expressed as (λ - λ₁)^{m₁}(λ - λ₂)^{m₂}... , where λ₁ and λ₂ are eigenvalues and m₁, m₂ are their respective exponents, then m₁ gives you the algebraic multiplicity for λ₁ and m₂ gives you it for λ₂.
• Evaluate how knowing the algebraic multiplicity of an eigenvalue can affect our understanding of a matrix's diagonalizability.
  • Knowing the algebraic multiplicity of an eigenvalue is essential in evaluating whether a matrix is diagonalizable. If all eigenvalues have their algebraic multiplicities equal to their geometric multiplicities, then we can conclude that there are enough linearly independent eigenvectors to form a basis. This means we can represent the matrix in a diagonal form. However, if any eigenvalue's algebraic multiplicity exceeds its geometric multiplicity, it implies that not enough independent eigenvectors exist, making diagonalization impossible for that matrix.

Related terms

Eigenvalue: An eigenvalue is a scalar that indicates how much an eigenvector is stretched or compressed during a linear transformation represented by a matrix.

Characteristic Polynomial: The characteristic polynomial is a polynomial derived from a matrix that encapsulates its eigenvalues, typically expressed as det(A - λI) = 0, where A is the matrix, λ is an eigenvalue, and I is the identity matrix.

Geometric Multiplicity: Geometric multiplicity refers to the number of linearly independent eigenvectors associated with a given eigenvalue, indicating the dimensionality of its eigenspace.