33.10. Repeated Eigenvalues and Geometric Multiplicity

Let’s dig deeper into multiplicity:

• Algebraic multiplicity (AM): Number of times an eigenvalue appears in the characteristic equation.
• Geometric multiplicity (GM): Number of linearly independent eigenvectors associated with that eigenvalue.

This chunk introduces two key concepts related to eigenvalues: algebraic multiplicity and geometric multiplicity. The algebraic multiplicity (AM) of an eigenvalue is how many times that eigenvalue shows up as a solution in the characteristic equation of a matrix. For example, if a matrix has an eigenvalue of 2 that appears three times in its characteristic polynomial, then its AM is 3.

In contrast, the geometric multiplicity (GM) refers to the number of linearly independent eigenvectors corresponding to the same eigenvalue. To have full usefulness in many applications, the GM must equal the AM for a matrix to be diagonalizable. This means we need to have as many independent directions (eigenvectors) as there are repetitions of an eigenvalue.

Think of algebraic multiplicity as the number of times a song is played on the radio (it could be quite often), while geometric multiplicity is like the number of distinct radio stations that play that song. If a song is played on multiple stations and you count each occurrence (AM), but you want to find out how many stations are actually playing it independently (GM). You need a station for each play to ensure you can hear it from different angles or interpretations.

Key Rule: A matrix A is diagonalizable if GM = AM for all eigenvalues.

This key rule states that a matrix is diagonalizable only when the geometric multiplicity (GM) is equal to the algebraic multiplicity (AM) for each of its eigenvalues. This equality ensures that there are enough independent eigenvectors to form a complete basis for the vector space in which the matrix operates. Therefore, if you find an eigenvalue whose GM is less than its AM, you can immediately conclude that the matrix cannot be diagonalized.

Imagine trying to organize a group discussion. For every participant (an eigenvalue), you want to ensure that there are enough unique perspectives (independent eigenvectors). If one participant's views are repeated multiple times (high AM) but they can only bring one unique idea (low GM), then your discussion won't be as enriching, and you won't achieve a varied and comprehensive conversation (diagonalization).

Example of Non-Diagonalizable Matrix A= [2 1] [0 2]

• Characteristic polynomial: ¿
• Eigenvalue: λ=2, AM = 2 A−2I= [0 1] [0 0] Solving (A−2I)v=0 gives only one linearly independent eigenvector ⇒ GM = 1. So, A is not diagonalizable.

In this example, we're analyzing a specific matrix to determine its diagonalizability. The matrix has an eigenvalue, λ = 2, which appears twice in its characteristic polynomial (hence AM = 2). However, when we compute (A−2I) and solve for eigenvectors, we find that we can only extract one linearly independent eigenvector (GM = 1). This discrepancy between GM and AM indicates that we do not have sufficient independent eigenvectors to diagonalize the matrix, confirming that this matrix is not diagonalizable.

Consider a project team where only one person knows how to handle a specific task, even though there are multiple people assigned to that role. If those extra people can only echo what the original team member says (one independent voice), then while you have many 'helpers' (high AM), you lack enough unique insights or solutions (low GM) to lead effectively. As a result, your team struggles to adapt or delegate more effectively, just like the matrix that can't be diagonalized.

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