Algebraic and Geometric Multiplicity

Algebraic multiplicity is the number of times an eigenvalue appears as a root of a characteristic polynomial of a matrix. Whereas, geometric multiplicity is the number of linearly independent eigenvectors connected with that eigenvalue. Both concepts help in analyzing the properties and structure of matrices.

In this article, we will see the differences between the algebraic and geometric multiplicities their significance and how they are computed.

What is Algebraic Multiplicity?

The number of times an eigenvalue occurs as the root of the matrix's characteristic polynomial is known as its algebraic multiplicity.

Put more simply it is the number of times a specific eigenvalue appears in the polynomial equation that is produced by setting the matrix (A−λI) determinant to zero where A is the matrix, λ is the eigenvalue and I is the identity matrix.

For example, if the characteristic polynomial of the matrix is (x−2)3(x−4)2 the eigenvalue 2 has an algebraic multiplicity of the 3 and eigenvalue 4 has an algebraic multiplicity of the 2. The algebraic multiplicity indicates how many times each eigenvalue appears as a solution to the characteristic equation.

What is Geometric Multiplicity?

The Geometric multiplicity of the eigenvalue is defined as the dimension of eigenspace associated with that eigenvalue.

The eigenspace is the set of the all eigenvectors corresponding to the particular eigenvalue along with the zero vector. Essentially, the geometric multiplicity tells us the number of the linearly independent eigenvectors associated with the eigenvalue.

The Continuing with the previous example if the eigenvalue 2 has the two linearly independent eigenvectors then its geometric multiplicity is 2. If the eigenvalue 4 has one linearly independent eigenvector then its geometric multiplicity is 1. The Geometric multiplicity provides the insight into the nature of the eigenvectors and their span.

Calculating Multiplicities

Calculating multiplicities involves two parts:

• Finding Algebraic Multiplicity
• Finding Geometric Multiplicity

Finding Algebraic Multiplicity

To find the algebraic multiplicity of an eigenvalue:

• Compute the characteristic polynomial of the matrix.
• The Factor the polynomial.
• The algebraic multiplicity of the eigenvalue is the exponent of the corresponding factor.

Example: For matrix A:

Characteristic Polynomial =(λ−2)³(λ−3)

• Eigenvalue 2: Algebraic multiplicity = 3
• Eigenvalue 3: Algebraic multiplicity = 1

Finding Geometric Multiplicity

To find the geometric multiplicity of the eigenvalue:

• Solve the equation (A − λI)x = 0 to the find the eigenspace.
• The number of the linearly independent solutions is the geometric multiplicity.

Example: For eigenvalue 2 solve (A − 2I)x = 0. If there are 2 linearly independent solutions the geometric multiplicity is 2.

Read More about How to find Geometric Multiplicity?

Significance of Algebraic and Geometric Multiplicity

• The Algebraic Multiplicity provides information on the total number of the eigenvalues.
• The Geometric Multiplicity gives the number of the independent directions in the which the matrix acts as a stretching factor.
• Key Relationship: For any eigenvalue λ the geometric multiplicity is always less than or equal to the algebraic multiplicity.

Conclusion

Understanding algebraic and geometric multiplicities is fundamental in the linear algebra as they provide the insights into the properties of the matrices and the behavior of the linear transformations. By mastering these concepts students can better analyze and solve the linear systems paving the way for the more advanced studies in the mathematics and engineering.

Read More,

• Eigenvalues and Eigenvectors
• Applications of Eigenvalues and Eigenvectors
• What is the difference between eigenvalues and eigenvectors?

Solved Examples of Algebraic and Geometric Multiplicity

Q1: Given matrix B \begin{bmatrix} 4 & 1 \\ 2 & 3 \end{bmatrix} find the eigenvalues. The Determine the algebraic and geometric multiplicities.

Solution:

Characteristic Polynomial: det(B - λI) = det( ( 4 - λ 1 ) ( 2 3 - λ ) ) = (λ - 5)(λ - 2)

Eigenvalues: λ = 5, 2

Algebraic Multiplicity:

• For λ = 5: Algebraic multiplicity = 1
• For λ = 2: Algebraic multiplicity = 1

Geometric Multiplicity:

• For λ = 5: Solve (B - 5I)x = 0 one independent solution geometric multiplicity = 1
• For λ = 2: Solve (B - 2I)x = 0 one independent solution geometric multiplicity = 1

Q2: Matrix C \begin{bmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{bmatrix} find the eigenvalues. Determine the algebraic and geometric multiplicities.

Solution:

Characteristic Polynomial: det(C - λI) = (2 - λ)^2 (3 - λ)

Eigenvalues: λ = 2, 3

Algebraic Multiplicity:

• For λ = 2: Algebraic multiplicity = 2
• For λ = 3: Algebraic multiplicity = 1

Geometric Multiplicity:

• For λ = 2: Solve (C - 2I)x = 0 two independent solutions geometric multiplicity = 2
• For λ = 3: Solve (C - 3I)x = 0 one independent solution geometric multiplicity = 1

Q3: Matrix: C = \begin{pmatrix} 3 & 1 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 5 \end{pmatrix}

Solution:

Characteristic polynomial: \det(C - \lambda I) = \det \begin{pmatrix} 3 - \lambda & 1 & 0 \\ 0 & 3 - \lambda & 0 \\ 0 & 0 & 5 - \lambda \end{pmatrix} = (3 - \lambda)^2 (5 - \lambda)

Solve for eigenvalues:

(3 - \lambda)^2 (5 - \lambda) = 0

⇒ \lambda = 3, 5

Algebraic multiplicity:

\lambda = 3: Algebraic multiplicity is 2.

⇒ \lambda = 5: Algebraic multiplicity is 1.

Geometric multiplicity:

\lambda = 3: Solve (C - 3I)x = 0

⇒ \lambda = 5: Solve (C - 5I)x = 0

• \lambda = 3: Geometric multiplicity is 2.
• \lambda = 5: Geometric multiplicity is 1.

Q4: Matrix: D = \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix}

Solution:

Characteristic polynomial:

\det(D - \lambda I) = \det \begin{pmatrix} 2 - \lambda & 0 \\ 0 & 2 - \lambda \end{pmatrix} = (2 - \lambda)^2

Solve for eigenvalues:

(2 - \lambda)^2 = 0

\lambda = 2

Algebraic multiplicity:

\lambda = 2: Algebraic multiplicity is 2.

Geometric multiplicity:

Solve (D - 2I)x = 0

Geometric multiplicity is 2 (two linearly independent eigenvectors).

Q5: Matrix: E = \begin{pmatrix} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 3 \end{pmatrix}

Solution:

Characteristic polynomial:

\det(E - \lambda I) = \det \begin{pmatrix} 1 - \lambda & 1 & 0 \\ 0 & 1 - \lambda & 0 \\ 0 & 0 & 3 - \lambda \end{pmatrix} = (1 - \lambda)^2 (3 - \lambda)

Solve for eigenvalues:

(1 - \lambda)^2 (3 - \lambda) = 0

\lambda = 1, 3

Algebraic multiplicity:

\lambda = 1: Algebraic multiplicity is 2.

\lambda = 3: Algebraic multiplicity is 1.

Geometric multiplicity:

\lambda = 1: Solve (E - 1I)x = 0

For \lambda = 3: Solve (E - 3I)x = 0

\lambda = 1: Geometric multiplicity is 1.

\lambda = 3: Geometric multiplicity is 1.

Practice Questions

Q1: Matrix: F = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}

Find the algebraic and geometric multiplicities of the eigenvalues.

Q2: Matrix: G = \begin{pmatrix} 4 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{pmatrix}

Determine the algebraic and geometric multiplicities of the eigenvalues.

Q3: Matrix: H = \begin{pmatrix} 1 & 2 \\ 2 & 1 \end{pmatrix}

Calculate the algebraic and geometric multiplicities for the each eigenvalue.

Q4: Matrix: I = \begin{pmatrix} 3 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 3 \end{pmatrix}

Identify the algebraic and geometric multiplicities.

Q5: Matrix: J = \begin{pmatrix} 2 & 1 \\ 0 & 2 \end{pmatrix}

Find the algebraic and geometric multiplicities for the eigenvalues.

Q6: Matrix: K = \begin{pmatrix} 5 & 4 \\ 4 & 5 \end{pmatrix}

Determine the algebraic and geometric multiplicities for the eigenvalues.

Q7: Matrix: L = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{pmatrix}

Calculate the algebraic and geometric multiplicities for the each eigenvalue.

Q8: Matrix: M = \begin{pmatrix} 0 & 2 \\ 2 & 0 \end{pmatrix}

Identify the algebraic and geometric multiplicities.

Q9: Matrix: N = \begin{pmatrix} 3 & 1 \\ 0 & 3 \end{pmatrix}

Find the algebraic and geometric multiplicities for the eigenvalues.

Q10: Matrix: O = \begin{pmatrix} 2 & 0 \\ 0 & 5 \end{pmatrix}

Determine the algebraic and geometric multiplicities for the eigenvalues.