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14. Multiplicity

The eigenvalues of a matrix are the roots of its characteristic polynomial. Given what we know about polynomials, there are some conclusions it’s worth stating clearly.

Property 14.1 (Eigenvalue properties)

Suppose \(\bfA\) is an \(n\times n\) matrix. Then

  1. \(\bfA\) has at least one and at most \(n\) distinct complex eigenvalues, and

  2. If \(\bfA\) is real, then any complex eigenvalues occur in conjugate pairs, as do their associated eigenvectors.

In general, we can factor a characteristic polynomial \(p\) to get

\[p(\lambda) = (z-\lambda_1)^{m_1}(z-\lambda_2)^{m_2}\cdots(z-\lambda_k)^{m_k},\]

for positive integer exponents such that \(m_1+\cdots+m_k=n\). These exponents are the multiplicities of the roots, and that idea carries to the eigenvalues as well.

Definition 14.2 (Algebraic multiplicity)

The algebraic multiplicity of an eigenvalue is its multiplicity as a root of the characteristic polynomial.

14.1. Geometric multiplicity

The following example illustrates a possibility unique to eigenvalues of algebraic multiplicity greater than 1.

Example

Find the eigenspaces of \(\bfA=\twomat{4}{1}{0}{4}\).

Solution

The characteristic polynomial is

\[\twodet{4-\lambda}{1}{0}{4-\lambda} = (4-\lambda)^2,\]

so the double root \(\lambda_1=4\) is the only eigenvalue. Since

\[\bfA - 4\meye = \twomat{0}{1}{0}{0},\]

the eigenspace has basis \(\twovec{1}{0}\).

This leads us to define a second notion of multiplicity for an eigenvalue.

Definition 14.3 (Geometric multiplicity)

The geometric multiplicity of an eigenvalue is the number of basis vectors in its associated eigenspace.

Here is an important fact we won’t try to prove.

Property 14.4

The geometric multiplicity of an eigenvalue is at least one and less than or equal to its algebraic multiplicity.

14.2. Defectiveness

In the above example we found a lone eigenvalue \(\lambda_1=4\) of algebraic multiplicity 2 whose geometric multiplicity, we now see, is 1. The identity matrix is a different sort of example.

Example

The \(2\times 2\) identity matrix \(\meye\) has a lone eigenvalue \(\lambda_1=1\) of algebraic multiplicity 2. The system \((\meye - \meye)\bfv=\bfzero\) has an RREF that is the zero matrix, so there are two free variables and two basis vectors. Hence the geometric multiplicity of \(\lambda_1\) is also 2.

The distinction between these cases is significant enough to warrant yet another definition and name.

Definition 14.5 (Defectiveness)

An eigenvalue \(\lambda\) whose geometric multiplicity is strictly less than its algebraic multiplicity is said to be defective. A matrix is called defective if any of its eigenvalues are defective.

As we will see later on, defective matrices often complicate the application of eigenvalue analysis. They are rare in the sense that a random matrix has zero probability of being defective, but they do play a role. Since multiplicities are always at least one, there is a simple and common case in which we are certain that a matrix is not defective.

Theorem 14.6 (Distinct eigenvalues)

If \(\bfA\in\cmn{n}{n}\) has \(n\) distinct eigenvalues, then \(\bfA\) is not defective.

For \(n=2\), the possibilities in the case of algebraic multiplicity equal to 2 are easy to pin down even further.

Theorem 14.7 (\(2\times 2\) defectivenss)

Any \(\bfA\in\cmn{2}{2}\) that has a single repeated eigenvalue is either defective or a multiple of the identity matrix.

13. Eigenvalues 1. Overview