Eigenvalues and eigenvectors are crucial in understanding matrix transformations. Algebraic and geometric multiplicities provide deeper insights into these concepts, revealing important properties of matrices and their eigenvalues.

These multiplicities help determine if a matrix is diagonalizable, a key characteristic in linear algebra. Understanding their relationship is essential for analyzing matrix behavior and solving complex problems in various fields.

Algebraic vs Geometric Multiplicities

Definition and Comparison

• The algebraic multiplicity of an eigenvalue is the number of times the eigenvalue appears as a root of the characteristic polynomial
• The geometric multiplicity of an eigenvalue is the dimension of the eigenspace associated with that eigenvalue
  • Represents the number of linearly independent eigenvectors corresponding to the eigenvalue
• The algebraic multiplicity is always greater than or equal to the geometric multiplicity for a given eigenvalue
• If the algebraic and geometric multiplicities are equal for all eigenvalues, the matrix is diagonalizable

Implications for Matrix Diagonalization

• When the algebraic and geometric multiplicities are equal for all eigenvalues, the matrix is diagonalizable
  • Diagonalizable matrices can be transformed into a diagonal matrix using a change of basis formed by the eigenvectors
• If the algebraic multiplicity is strictly greater than the geometric multiplicity for at least one eigenvalue, the matrix is not diagonalizable
  • Non-diagonalizable matrices, also known as defective matrices, cannot be transformed into a diagonal matrix using eigenvectors as a basis
• The sum of the geometric multiplicities of all eigenvalues is equal to the dimension of the matrix

Geometric Multiplicity of Eigenvalues

Determining the Geometric Multiplicity

• To find the geometric multiplicity, first determine the eigenspace associated with the eigenvalue by solving the equation $(A - λI)v = 0$
  • $A$ is the matrix, $λ$ is the eigenvalue, $I$ is the identity matrix, and $v$ is the eigenvector
• The geometric multiplicity is the dimension of the nullspace (or kernel) of the matrix $A - λI$
  • Compute the rank of $A - λI$ and subtract it from the number of columns (or rows) of the matrix
• The geometric multiplicity can also be determined by finding the number of free variables in the solution of the equation $(A - λI)v = 0$
  • Free variables represent the dimension of the solution space, which is the geometric multiplicity

Examples and Techniques

• To find the geometric multiplicity of an eigenvalue $λ$, set up the equation $(A - λI)v = 0$ and solve for the eigenvector $v$
  • The number of linearly independent solutions (eigenvectors) is the geometric multiplicity
• Gaussian elimination can be used to find the rank of the matrix $A - λI$, which helps determine the geometric multiplicity
  • The rank is the number of non-zero rows after Gaussian elimination
  • The geometric multiplicity is the number of columns (or rows) minus the rank
• The geometric multiplicity can also be found by analyzing the reduced row echelon form of the matrix $A - λI$
  • The number of free variables (columns with no leading 1s) is equal to the geometric multiplicity

Multiplicity Relationships

Algebraic and Geometric Multiplicity Inequality

• The algebraic multiplicity is always greater than or equal to the geometric multiplicity for a given eigenvalue
  • This relationship holds for all matrices, whether diagonalizable or not
• If the algebraic and geometric multiplicities are equal for all eigenvalues, the matrix is diagonalizable
  • Diagonalizable matrices have a full set of linearly independent eigenvectors
• If the algebraic multiplicity is strictly greater than the geometric multiplicity for at least one eigenvalue, the matrix is not diagonalizable
  • Non-diagonalizable matrices, also known as defective matrices, do not have a full set of linearly independent eigenvectors

Sum of Geometric Multiplicities

• The sum of the geometric multiplicities of all eigenvalues is equal to the dimension of the matrix
  • This relationship holds for all square matrices, regardless of diagonalizability
• The dimension of the matrix is the number of rows (or columns) in the square matrix
• The sum of the algebraic multiplicities of all eigenvalues is also equal to the dimension of the matrix
  • This is because the algebraic multiplicities represent the roots of the characteristic polynomial, which has a degree equal to the matrix dimension

Equal vs Different Multiplicities

Diagonalizable Matrices

• If a matrix is diagonalizable, the algebraic and geometric multiplicities are equal for all eigenvalues
  • Diagonalizable matrices have a full set of linearly independent eigenvectors
• If a matrix has distinct eigenvalues, the algebraic and geometric multiplicities are always equal
  • Distinct eigenvalues guarantee linear independence of the corresponding eigenvectors
• Matrices with repeated eigenvalues may have equal algebraic and geometric multiplicities if the corresponding eigenvectors are linearly independent
  • The Jordan canonical form can be used to analyze matrices with repeated eigenvalues

Non-Diagonalizable Matrices

• If a matrix is not diagonalizable, there exists at least one eigenvalue for which the algebraic multiplicity is strictly greater than the geometric multiplicity
  • Non-diagonalizable matrices, also known as defective matrices, do not have a full set of linearly independent eigenvectors
• Defective matrices always have at least one eigenvalue with algebraic multiplicity greater than its geometric multiplicity
  • The Jordan canonical form of a defective matrix contains Jordan blocks with size greater than 1 for these eigenvalues
• Matrices with repeated eigenvalues may have different algebraic and geometric multiplicities if the corresponding eigenvectors are not linearly independent
  • The generalized eigenvectors can be used to analyze defective matrices and construct the Jordan canonical form