Algebraic And Geometric Multiplicity

Algebraic and Geometric Multiplicity: Understanding Eigenvalues Deeply

Eigenvalues and eigenvectors are fundamental concepts in linear algebra, offering crucial insights into the behavior of linear transformations. Understanding eigenvalues allows us to analyze matrices and their transformations effectively. Central to this understanding are two related concepts: algebraic multiplicity and geometric multiplicity. These values describe the "size" of an eigenvalue in different ways, and their relationship reveals important information about the structure of the associated linear transformation. This article will explore both concepts, detailing their definitions, calculations, and significance.

1. Eigenvalues and Eigenvectors: A Quick Recap

Before diving into algebraic and geometric multiplicity, let's briefly revisit the core concepts of eigenvalues and eigenvectors. Consider a square matrix A. A non-zero vector 'v' is an eigenvector of A if multiplying A by v results in a scalar multiple of v:

Av = λv

where λ is a scalar called the eigenvalue corresponding to the eigenvector v. Essentially, the eigenvector's direction remains unchanged under the transformation represented by A; only its length is scaled by the eigenvalue. Finding eigenvalues involves solving the characteristic equation:

det(A - λI) = 0

where I is the identity matrix. The solutions to this equation are the eigenvalues of A.

2. Defining Algebraic Multiplicity

The algebraic multiplicity of an eigenvalue λ is the multiplicity of λ as a root of the characteristic polynomial, det(A - λI). In simpler terms, it's how many times λ appears as a solution to the characteristic equation. If the characteristic polynomial factors as (λ - λ₁)^m₁ (λ - λ₂)^m₂ … (λ - λₖ)^mₖ, then the algebraic multiplicity of λᵢ is mᵢ.

Example:

Consider the matrix A = [[2, 0], [0, 2]]. The characteristic equation is (2-λ)² = 0. This has a single root, λ = 2, with an algebraic multiplicity of 2.

3. Defining Geometric Multiplicity

The geometric multiplicity of an eigenvalue λ is the dimension of the eigenspace associated with λ. The eigenspace is the set of all eigenvectors corresponding to λ, along with the zero vector. It's essentially the null space of the matrix (A - λI). Geometric multiplicity represents the number of linearly independent eigenvectors associated with the eigenvalue. It can be determined by finding the nullity (dimension of the null space) of (A - λI). This is equivalent to finding the number of free variables in the reduced row echelon form of (A - λI).

Example:

For the same matrix A = [[2, 0], [0, 2]], the eigenspace for λ = 2 is the set of all vectors of the form [x, y] where 2x = 2x and 2y = 2y (this comes from solving (A - 2I)v = 0). This simplifies to all vectors in R². Therefore, the geometric multiplicity of λ = 2 is 2 (as there are two linearly independent vectors spanning this space, e.g., [1,0] and [0,1]).

4. Relationship Between Algebraic and Geometric Multiplicity

The geometric multiplicity of an eigenvalue is always less than or equal to its algebraic multiplicity:

1 ≤ Geometric Multiplicity (λ) ≤ Algebraic Multiplicity (λ)

When the geometric multiplicity equals the algebraic multiplicity for all eigenvalues, the matrix is diagonalizable. This means we can find a matrix P such that P⁻¹AP is a diagonal matrix with the eigenvalues on the diagonal. If the geometric multiplicity is less than the algebraic multiplicity for at least one eigenvalue, the matrix is not diagonalizable. This often indicates the presence of repeated eigenvalues and a shortage of linearly independent eigenvectors.

5. Consequences of Multiplicity Discrepancy

A discrepancy between algebraic and geometric multiplicity has significant implications. It indicates that the linear transformation represented by the matrix is more complex and cannot be fully understood simply by its eigenvalues. It signifies that the transformation has a non-trivial Jordan canonical form, a more general representation than a diagonal matrix that accounts for the limitations in finding a sufficient number of linearly independent eigenvectors. This leads to difficulties in solving certain types of linear systems and in applying spectral decompositions.

Summary

Algebraic and geometric multiplicity are essential concepts for characterizing eigenvalues and understanding the structure of linear transformations. The algebraic multiplicity reflects the multiplicity of an eigenvalue as a root of the characteristic polynomial, while the geometric multiplicity represents the dimension of the corresponding eigenspace. The relationship between these two multiplicities – geometric multiplicity always being less than or equal to algebraic multiplicity – determines the diagonalizability of a matrix. A mismatch indicates a more complex transformation that requires more sophisticated techniques for analysis.

FAQs

1. Q: Why is the geometric multiplicity always less than or equal to the algebraic multiplicity?
A: The geometric multiplicity represents the number of linearly independent eigenvectors associated with an eigenvalue. The algebraic multiplicity represents how many times the eigenvalue appears as a root of the characteristic equation. Since linearly independent eigenvectors are required for a basis of the eigenspace, you can never have more linearly independent eigenvectors than the algebraic multiplicity indicates.

2. Q: What does it mean if a matrix is not diagonalizable?
A: A non-diagonalizable matrix implies that the geometric multiplicity of at least one eigenvalue is less than its algebraic multiplicity. This means we can't find a sufficient number of linearly independent eigenvectors to form a basis for the entire vector space, and a simple diagonal representation is not possible. A Jordan canonical form is needed instead.

3. Q: How do I calculate geometric multiplicity?
A: Calculate (A - λI), where A is the matrix and λ is the eigenvalue. Then find the null space of (A - λI) by performing Gaussian elimination or other equivalent methods. The dimension of the null space is the geometric multiplicity.

4. Q: What is the significance of diagonalizability?
A: Diagonalizable matrices significantly simplify many linear algebra operations. They allow for easier calculations of matrix powers, solutions to linear systems, and applications in areas like Markov chains and differential equations.

5. Q: Can the algebraic multiplicity be zero?
A: No, if λ is an eigenvalue, its algebraic multiplicity must be at least 1, as it is a root of the characteristic polynomial. A zero algebraic multiplicity would mean λ is not an eigenvalue.

Search Results:

What is the geometric intuition behind algebraic multiplicity? 3 Mar 2018 · The algebraic multiplicity of an eigenvalue $\lambda$ is the number of times $\lambda$ appears as a root of the characteristic polynomial. The geometric multiplicity of an eigenvalue $\lambda$ is dimension of the eigenspace of the eigenvalue $\lambda$.

linear algebra - How to find the multiplicity of eigenvalues ... The dimension of this kernel is then said to be the geometric multiplicity of the eigen-value. Hence, in one case, one has to compute some polynomial; while, on the other hand, one has to compute some transformations, to find its kernel, and to determine the dimension of the kernel, to find the multiplicites of eigen-values.

Examples for proof of geometric vs. algebraic multiplicity Here you see a supposedly easy proof of a well-known theorem in linear algebra: Although I know I should understand this, I don't :-( Obviously there are too many indices and stuff, so I don't se...

Algebraic and geometric multiplicities of eigenvalues of a 26 Oct 2017 · The geometric multiplicity is the dimension of the eigenspace of each eigenvalue and the algebraic multiplicity is the number of times the eigenvalue appears in the factorization of the caracteristic polynomial. In your example the algebraic multiplicity of 3 is 1 and this implies that its geometric multiplicity is also 1.

linear algebra - Algebraic multiplicity = geometric multiplicity ... 24 Jun 2016 · And if you mean the usual definition of diagonalizability, then its algebraic and geometric multiplicity coincide. $\endgroup$ – cjackal Commented Jun 24, 2016 at 7:29

linear algebra - Can someone explain geometric multiplicity ... 25 Feb 2014 · $\begingroup$ Geometric multiplicity, as you say, is the number of linearly independent eigenvectors related to a given eigenvalue. Whereas the algebraic multiplicity is the dimension of the invariant subspace. In general $1\le$ geometric $\le$ algebraic. Think for example of the basic 2 times 2 nilpotent matrix.

why geometric multiplicity is bounded by algebraic multiplicity? The geometric multiplicity the be the dimension of the eigenspace associated with the eigenvalue $\lambda_i$. For example: $\begin{bmatrix}1&1\\0&1\end{bmatrix}$ has root $1$ with algebraic multiplicity $2$, but the geometric multiplicity $1$. My Question: Why is the geometric multiplicity always bounded by algebraic multiplicity? Thanks.

What are the relations between geometric multiplicity and … 15 Apr 2018 · $\begingroup$ The linear transformation is diagonalizable if and only if the geometric multiplicity of each eigenvalue is equal to its algebraic multiplicity. (This follows from what you actually said.) $\endgroup$ –

linear algebra - For a symmetric matrix, the geometric and … We can always construct an Eigenspace for each $\lambda$ with size of Algebraic Multiplicity $\mu(\lambda)$. For a specific eigenvalue $\lambda$, if Geometric Multiplicity $\gamma(\lambda)$ is equal to Algebraic Multiplicity $\mu(\lambda)$, this means the size of the largest Jordan Block should be 1 and there are $\mu=\gamma$ blocks for $\lambda$.

how to Obtain the algebraic and geometric multiplicity of each ... 16 Feb 2020 · Learn more about matrices, eigenvalue, eigenvector, algebraic and geometric multiplicity MATLAB. Matlab code .

Algebraic And Geometric Multiplicity