To solve the question:

Definitions:

1. Algebraic Multiplicity: This is the number of times an eigenvalue $\lambda$ appears as a root of the characteristic polynomial.
2. Geometric Multiplicity: This is the number of linearly independent eigenvectors corresponding to the eigenvalue $\lambda$, which is equal to the dimension of the eigenspace associated with $\lambda$.

Key Property:

The geometric multiplicity of an eigenvalue $\lambda$ (i.e., the minimum number of linearly independent eigenvectors) is always less than or equal to its algebraic multiplicity.

Given:

• $\lambda$ has algebraic multiplicity $3$.

Answer:

The minimum number of linearly independent eigenvectors is 1. This occurs when the geometric multiplicity is at its minimum value, which is $1$. This happens if the eigenspace is 1-dimensional.

Let me know if you'd like a detailed breakdown or further clarifications.

Expansion Questions:

1. What is the geometric multiplicity if the eigenspace has more than one independent eigenvector for $\lambda$?
2. How is the characteristic polynomial related to the eigenvalues and their multiplicities?
3. Can a matrix be diagonalizable if the geometric multiplicity of an eigenvalue is less than its algebraic multiplicity?
4. What is the maximum geometric multiplicity for an eigenvalue of a matrix with algebraic multiplicity $m$?
5. How do Jordan forms represent eigenvalues and eigenvectors of a matrix?

Tip:

For a matrix to be diagonalizable, the sum of the geometric multiplicities of all eigenvalues must equal the size of the matrix.