Introduction to Eigenvectors and Eigenvalues

Understanding the concept of eigenvalues and eigenvectors is fundamental to linear algebra. A square matrix (A) applied to its eigenvector (v) will result in a scalar multiple of (v), i.e., (Av lambda v), where (lambda) is the eigenvalue associated with (v). However, the discussion narrows down to the number of linearly independent eigenvectors a matrix can have. This article delves into the intricacies that determine the number of such eigenvectors based on the scalar field and explores invariant subspaces as a more comprehensive approach.

Dependence on the Scalar Field

The existence and count of linearly independent eigenvectors for a given square matrix strongly depends on the scalar field over which the matrix is defined. Consider the real numbers, for instance. In this context, it is essential to understand the nuances:

Uncountably Many Eigenvectors for One Eigenvalue

For real matrices which have at least one eigenvalue, the set of all eigenvectors corresponding to a given eigenvalue spans a subspace. This subspace can be infinite-dimensional, meaning there are uncountably many linearly independent eigenvectors. This is a direct result of the vector space being Euclidean and having a continuum of real numbers.

Matrices with No Eigenvalues

However, not all matrices are lucky to have eigenvalues. There exist real matrices that do not have any eigenvalues. The classic example is a matrix that does not change under any rotation or scaling, such as a matrix with a complex eigenvalue whose real part is zero. Such matrices do not have real eigenvectors.

Eigenvectors and Linear Transformations

Understanding eigenvectors through the lens of scalar multiples is crucial. Given an eigenvector (v), any nonzero scalar multiple of (v) is also an eigenvector, associated with the same eigenvalue (lambda). This property indicates that the direction of the eigenvector is conserved under the transformation defined by the matrix, merely scaled by the eigenvalue.

Linearly Independent Eigenvectors

The concept of linearly independent eigenvectors is pivotal in the study of linear transformations. A set of eigenvectors is said to be linearly independent if no eigenvector in the set can be expressed as a linear combination of the others. This linear independence ensures that the eigenvectors form a basis for the eigenspace associated with their respective eigenvalues.

The Wrong Question: Invariant Subspaces

While the question of the number of linearly independent eigenvectors is valid but not always the most insightful, another more comprehensive approach is to focus on invariant subspaces. An invariant subspace under a linear transformation is a subspace that remains fixed under the transformation. This transformation mapping from the space to itself will not change the elements of the subspace.

Invariant Subspaces and Minimal Polynomials

For each irreducible factor of the minimal polynomial of a matrix, there is an associated invariant subspace. The minimal polynomial is the monic polynomial of least degree that annihilates the matrix, i.e., (p(A) 0). The factorization of the minimal polynomial provides a decomposition of the vector space into a direct sum of these invariant subspaces. Each irreducible factor corresponds to a distinct invariant subspace, guaranteeing that the entire vector space can be decomposed into a sum of such invariant subspaces.

Example of Invariant Subspaces

Consider a matrix (A) with minimal polynomial (p(x) (x - lambda_1)(x - lambda_2)^2). The invariant subspaces here would correspond to the eigenspace for (lambda_1) and the generalized eigenspace for (lambda_2). The eigenspace is the set of eigenvectors for (lambda_1), and the generalized eigenspace includes both eigenvectors and generalized eigenvectors for (lambda_2). Together, these subspaces form a complete decomposition of the vector space.