Eigenspaces

When a linear transformation acts on a vector, it typically changes both its magnitude and direction. However, eigenvectors are special: they only change in magnitude, not direction. Each eigenvector is associated with a specific eigenvalue, which quantifies this scaling. An eigenspace takes this concept further, grouping all such special vectors that share the same scaling factor (eigenvalue) into a single geometric subspace. Understanding eigenspaces helps us visualize the invariant directions of a transformation.

An eigenspace is the set of all eigenvectors corresponding to a particular eigenvalue $\lambda$, along with the zero vector. For a given $n \times n$ matrix $A$ and an eigenvalue $\lambda$, the eigenspace $E_{\lambda}$ is defined as the set of all vectors $v$ such that $Av = \lambda v$. This collection forms a vector subspace of $\mathbb{R}^n$. Including the zero vector ensures that the set satisfies the properties of a vector subspace, even though the zero vector itself is not considered an eigenvector by definition.

The defining equation for an eigenvector, $Av = \lambda v$, can be rearranged to reveal its connection to a null space. Subtracting $\lambda v$ from both sides gives $Av - \lambda v = 0$. Since $v$ is a vector, we can factor it out by introducing the identity matrix $I$, resulting in $(A - \lambda I)v = 0$. This means that the eigenvectors corresponding to $\lambda$ are precisely the non-zero vectors in the null space of the matrix $(A - \lambda I)$. The eigenspace $E_{\lambda}$ is thus equivalent to the null space of $(A - \lambda I)$.

To find the basis for an eigenspace $E_{\lambda}$, we need to solve the homogeneous system $(A - \lambda I)v = 0$. This involves finding the null space of the matrix $(A - \lambda I)$. Typically, this is done by performing Gaussian elimination on $(A - \lambda I)$ to reduce it to row echelon form. The free variables in the resulting system will correspond to the basis vectors of the null space, which are the basis vectors for the eigenspace. These basis vectors are linearly independent and span the entire eigenspace.

The dimension of an eigenspace $E_{\lambda}$ is the number of linearly independent eigenvectors associated with the eigenvalue $\lambda$. This dimension is also known as the geometric multiplicity of the eigenvalue. It is equal to the number of vectors in any basis for $E_{\lambda}$. A geometric multiplicity of 1 means the eigenspace is a line, while a geometric multiplicity of 2 means it's a plane, and so on. The geometric multiplicity is always less than or equal to the algebraic multiplicity (the number of times an eigenvalue appears as a root of the characteristic polynomial).

When an eigenvalue has an algebraic multiplicity greater than 1, its corresponding eigenspace can have a geometric multiplicity that is either equal to or less than the algebraic multiplicity. If the geometric multiplicity is less than the algebraic multiplicity, the matrix is said to be defective and cannot be diagonalized. This scenario often arises in transformations that involve shearing or rotations that don't fully align with the axes. For example, a $2 \times 2$ matrix might have a single eigenvalue with algebraic multiplicity 2, but its eigenspace might only be a 1-dimensional line (geometric multiplicity 1).