The dimension of the eigenspace is 1.
The dimension of an eigenspace is fundamentally determined by the number of linearly independent eigenvectors associated with a specific eigenvalue. When this dimension is 1, it signifies that there is only one unique direction (or line) along which eigenvectors can exist for that particular eigenvalue.
This means that while there might be infinitely many eigenvectors for a given eigenvalue (as any non-zero scalar multiple of an eigenvector is also an eigenvector), they all lie along the same line in the vector space, thus representing only one linearly independent direction.
Understanding Eigenspace Dimensions
An eigenspace is a fundamental concept in linear algebra, representing the set of all eigenvectors corresponding to a particular eigenvalue, along with the zero vector. Its dimension, also known as the geometric multiplicity of the eigenvalue, reveals the 'size' of this space and the number of independent directions defined by the eigenvectors.
- Definition: The dimension of an eigenspace is equivalent to the number of linearly independent eigenvectors that can be found for a specific eigenvalue.
- Calculation: It is mathematically determined by the nullity of the matrix $(A - \lambda I)$, where $A$ is the original matrix, $\lambda$ is the eigenvalue, and $I$ is the identity matrix. This nullity represents the dimension of the null space of $(A - \lambda I)$, which is precisely the eigenspace for $\lambda$.
The table below illustrates the implications of different eigenspace dimensions:
| Eigenspace Dimension | Implication for Eigenvectors |
|---|---|
| 1 | One unique direction for eigenvectors (a line of solutions) |
| > 1 | Multiple unique directions (a plane or higher-dimensional space of solutions) |
Practical Implications of a 1-Dimensional Eigenspace
A dimension of 1 for an eigenspace simplifies the understanding of the transformation's effect on vectors within that space.
- Directional Consistency: All eigenvectors for that eigenvalue point along the same line. This means the linear transformation scales them (stretches or shrinks) without changing their fundamental direction (or reversing it).
- Basis Simplicity: A single non-zero eigenvector is sufficient to form a basis for this 1-dimensional eigenspace, providing the simplest representation of the characteristic direction.
- Limited Variability: Unlike higher-dimensional eigenspaces, there's no "plane" or "volume" of eigenvectors; just a single, straight line. This can be crucial in fields like dynamical systems, where unique directions often indicate stable or unstable modes of behavior.
