1 Definition and intuition

1.1 Multiplicity of polynomial roots

For a polynomial \(p(x)\), a number \(r\) is a root with *multiplicity* \(m\) if \((x-r)^m\) divides the polynomial, but \((x-r)^{m+1}\) does not. Intuitively, multiplicity measures how strongly the graph “touches” the axis at \(x=r\): a simple root crosses the axis, while higher multiplicity roots cause the graph to flatten and re-touch.

1.2 Algebraic multiplicity for eigenvalues

Given a square matrix \(A\), the *algebraic multiplicity* of an eigenvalue \(\lambda\) is the multiplicity of \(\lambda\) as a root of the characteristic polynomial \(\chi_A(t)=\det(tI-A)\). Thus, it counts how many times \(\lambda\) appears among the eigenvalues when counted with repetition according to \(\chi_A\).

1.3 Relationship to counting roots with repetition

Algebraic multiplicity formalizes a counting rule: if the characteristic polynomial factors as \[ \chi_A(t) = \prod_{i}(t-\lambda_i)^{m_i}, \] then each eigenvalue \(\lambda_i\) contributes \(m_i\) copies. This repetition is crucial because it influences canonical forms and the behavior of iterated or differentiated systems.

2 Algebraic multiplicity in polynomials

2.1 Factorization and repeated factors

2.1.1 Root multiplicity via factor powers

If a polynomial admits a factorization \[ p(x)=c\prod_{i}(x-r_i)^{m_i}, \] then the multiplicity of \(r_i\) as a root is exactly \(m_i\). This perspective is often the most direct: one reads multiplicity from the exponents in the factorization.

2.1.2 Characterization using divisibility

Multiplicity can be defined without fully factoring the polynomial. The integer \(m\ge 1\) is the multiplicity of \(r\) precisely when:

  • \((x-r)^m\) divides \(p(x)\), and
  • \((x-r)^{m+1}\) does not.

This divisibility criterion is central in algebraic proofs and in algorithmic computations that avoid complete factorization.

2.2 Using the derivative test

2.2.1 Multiple roots and common factors with derivatives

A standard criterion connects multiplicity to derivatives. Over a field of characteristic zero (or appropriate conditions), \(r\) is a root of multiplicity at least \(m\) if and only if \[ p(r)=p'(r)=p''(r)=\cdots=p^{(m-1)}(r)=0. \] Equivalently, if \(r\) has multiplicity \(m\), then \(p\) and its derivatives share corresponding factors near \(r\). In particular, a simple test is: \(r\) is a multiple root exactly when \(p(r)=0\) and \(p'(r)=0\).

2.3 Examples and worked calculations

2.3.1 Quadratic and cubic examples

  1. Quadratic:

\[ p(x)=(x-2)^2(x+1). \] The root \(x=2\) has algebraic multiplicity \(2\); the root \(x=-1\) has multiplicity \(1\).

  1. Cubic:

\[ p(x)=(x-3)^3. \] Here \(x=3\) has multiplicity \(3\). If one expands and differentiates, both \(p(3)=0\) and \(p'(3)=0\) hold, consistent with being a triple root.

2.3.2 Illustrative graphs of multiplicity

Graphically, multiplicity changes the local shape:

  • Odd multiplicity: the curve crosses the \(x\)-axis.
  • Even multiplicity: the curve touches the axis and turns back.
  • Higher multiplicity: the curve flattens more strongly at the root.

This matches the derivative criterion: more derivatives vanish at the root, indicating stronger tangency.

3 Algebraic multiplicity for matrices

3.1 Characteristic polynomial

For an \(n\times n\) matrix \(A\), the characteristic polynomial is \[ \chi_A(t)=\det(tI-A), \] a degree-\(n\) polynomial whose roots are the eigenvalues of \(A\), counted with algebraic multiplicity. The coefficients are tied to matrix invariants such as trace and determinant (and intermediate symmetric polynomials of eigenvalues).

3.2 Eigenvalues and their algebraic multiplicities

If \(\lambda\) is an eigenvalue, then \((t-\lambda)^{m}\) divides \(\chi_A(t)\), where \(m\) is the algebraic multiplicity. The sum of algebraic multiplicities of all eigenvalues equals \(n\), the matrix size.

3.3 Computing multiplicity from factorization

3.3.1 Diagonalizable vs. non-diagonalizable examples

  • Diagonalizable case: If \(A\) is diagonalizable over the field, each eigenvalue’s algebraic multiplicity equals the number of corresponding independent eigenvectors, but only when geometric multiplicity matches it. One can still have repeated eigenvalues: diagonalizability is about having enough eigenvectors, not about multiplicity being one.
  • Non-diagonalizable case: A repeated eigenvalue can have geometric multiplicity strictly smaller than algebraic multiplicity. Then the characteristic polynomial contains \((t-\lambda)^m\) with \(m> \) geometric multiplicity, and Jordan blocks emerge in the Jordan form.

3.4 Trace and determinant constraints

3.4.1 Connection to sums and products of eigenvalues

Even when multiplicities are not explicitly computed, they influence invariants:

  • The trace satisfies \(\mathrm{tr}(A)=\sum \lambda_i\), where each eigenvalue \(\lambda_i\) is counted with algebraic multiplicity.
  • The determinant satisfies \(\det(A)=\prod \lambda_i\), again with multiplicity.

These relationships act as consistency checks when eigenvalues (and their repetitions) are inferred.

4 Algebraic vs. geometric multiplicity

4.1 Eigenspace dimension and geometric multiplicity

For eigenvalue \(\lambda\), the eigenspace is \(\ker(A-\lambda I)\). Its dimension is the geometric multiplicity, denoted \(\dim \ker(A-\lambda I)\). While algebraic multiplicity counts occurrences in the characteristic polynomial, geometric multiplicity counts independent eigenvectors.

4.2 Fundamental inequality

A key theorem states: \[ \text{geometric multiplicity} \le \text{algebraic multiplicity}. \] So, geometric multiplicity can never exceed what the characteristic polynomial allows. When equality holds for every eigenvalue, the matrix is diagonalizable (under usual field conditions).

4.3 Implications for Jordan structure

4.3.1 Jordan blocks and multiplicity distributions

Jordan canonical form organizes repeated eigenvalues into blocks. For a fixed eigenvalue \(\lambda\), the algebraic multiplicity \(m\) equals the total size of all Jordan blocks associated with \(\lambda\). The geometric multiplicity equals the number of Jordan blocks for \(\lambda\). Consequently:

  • If algebraic multiplicity exceeds geometric multiplicity, at least one Jordan block has size greater than \(1\).
  • Larger algebraic multiplicity allows longer chains of generalized eigenvectors, reflecting stronger constraints on diagonalization.

5 Applications in linear algebra

5.1 Classification of linear transformations

Algebraic multiplicities help classify linear transformations by describing how eigenvalues repeat. Combined with geometric multiplicities, they determine whether the transformation can be fully decomposed into independent eigendirections or whether generalized eigenvectors are required.

5.2 Solving linear differential systems (conceptual)

For systems like \(\dot{x}=Ax\), solutions involve exponentials tied to eigenvalues. Repeated eigenvalues and their algebraic multiplicities can produce polynomial factors multiplying \(e^{\lambda t}\). These polynomial terms arise precisely when Jordan blocks are present, reflecting that algebraic multiplicity can exceed geometric multiplicity.

5.3 Stability and behavior under iteration (overview)

In discrete-time iteration \(x_{k+1}=Ax_k\), long-term behavior depends on eigenvalues and their repetitions. Algebraic multiplicity influences how transient growth manifests when eigenvalues are defective or repeated. Even when the dominant eigenvalue governs asymptotic scaling, multiplicity can determine the presence of additional polynomial growth/decay factors.

6 Common theorems and computational tools

6.1 Cayley–Hamilton perspective

The Cayley–Hamilton theorem states that \(A\) satisfies its characteristic polynomial: \[ \chi_A(A)=0. \] Algebraic multiplicities shape \(\chi_A\), and thus shape the annihilating polynomial relation satisfied by \(A\). This is often used to derive identities and to compute powers of \(A\) without diagonalization.

6.2 Minimal polynomial vs. characteristic polynomial

The minimal polynomial \(\mu_A(t)\) is the monic polynomial of least degree with \(\mu_A(A)=0\). For each eigenvalue \(\lambda\), the exponent of \((t-\lambda)\) in \(\mu_A\) is at most the exponent in \(\chi_A\), hence at most the algebraic multiplicity. When \(\mu_A\) has lower exponents, it indicates limited size of Jordan blocks.

6.3 Rank and nullity considerations

6.3.1 How multiplicity affects solution spaces

For systems \((A-\lambda I)x=0\), the solution space is the eigenspace, whose dimension equals geometric multiplicity. While algebraic multiplicity measures repetition in \(\chi_A\), rank-nullity ties the geometric multiplicity to the dimension of the nullspace: \[ \dim\ker(A-\lambda I)=n-\mathrm{rank}(A-\lambda I). \] If algebraic multiplicity is high but geometric multiplicity is low, then the eigenspace is small and generalized eigenvectors contribute to constructing solutions to related inhomogeneous or differential equations.

7 Extended notes and pitfalls

7.1 Distinguishing multiplicity from distinctness

A common confusion is to treat repeated eigenvalues as meaning “more eigenvectors.” Repetition affects the characteristic polynomial (algebraic multiplicity), but independent eigenvectors are counted by geometric multiplicity. Therefore, two matrices can share the same algebraic multiplicities yet differ in diagonalizability.

7.2 Handling repeated eigenvalues numerically

In numerical computations, eigenvalues that are theoretically repeated may appear as nearly equal clusters due to rounding and conditioning. Estimating algebraic multiplicity from computed eigenvalues can be misleading; stability analysis and residual checks (e.g., verifying eigenpair accuracy) are often required. For rigorous multiplicity, one typically relies on exact arithmetic or algebraic factorization where feasible.

7.3 Common misconceptions and clarifying examples

  • “Multiplicity means number of eigenvectors.” False: geometric multiplicity measures eigenvectors; algebraic multiplicity measures polynomial repetition.
  • “If an eigenvalue repeats, the matrix is automatically non-diagonalizable.” Not always. Repetition can coexist with diagonalizability when enough eigenvectors exist.
  • “Derivative test always works over any field.” The derivative criterion is reliable under standard assumptions (e.g., characteristic zero). In fields of positive characteristic, behavior can differ and requires separate treatment.

8 Mini glossary

8.1 Root, factor, and multiplicity

A root is a value that makes a polynomial vanish. A factor is a polynomial expression whose product forms the original polynomial. Multiplicity is the exponent of a corresponding factor, equivalently the number of times the root is counted in the factorization.

8.2 Eigenvalues, characteristic polynomial, eigenspace

An eigenvalue \(\lambda\) is a scalar for which \(A-\lambda I\) is singular. The characteristic polynomial \(\chi_A(t)=\det(tI-A)\) encodes eigenvalues and their algebraic multiplicities. The eigenspace is \(\ker(A-\lambda I)\), whose dimension is the geometric multiplicity.

8.3 Jordan form terminology

A Jordan block is a canonical block matrix associated with an eigenvalue that reflects the presence of generalized eigenvectors. For a given eigenvalue, the sum of the sizes of its Jordan blocks equals its algebraic multiplicity, while the number of Jordan blocks equals its geometric multiplicity.