How To Find Basis For Eigenspace

How to Find a Basis for an Eigenspace

Finding a basis for an eigenspace is a crucial step in linear algebra, particularly when dealing with eigenvalues and eigenvectors. This process allows us to understand the structure of a linear transformation and its effect on the vector space. This article will guide you through the steps, providing clear examples and explanations to solidify your understanding. Understanding eigenspaces is fundamental to many applications, including matrix diagonalization, solving systems of differential equations, and understanding the behavior of dynamic systems.

What is an Eigenspace?

Before diving into the process, let's define what an eigenspace is. An eigenspace corresponding to an eigenvalue λ of a matrix A is the set of all eigenvectors associated with that eigenvalue, along with the zero vector. In simpler terms, it's the subspace spanned by the eigenvectors associated with a particular eigenvalue.

Steps to Find a Basis for an Eigenspace:

1. Find the Eigenvalues: The first step is to determine the eigenvalues of the matrix A. This involves solving the characteristic equation, det(A - λI) = 0, where I is the identity matrix. This equation will yield a set of eigenvalues, denoted as λ₁, λ₂, etc.

2. For Each Eigenvalue, Solve the System (A - λI)x = 0: For each eigenvalue λᵢ you found in step 1, you need to solve the homogeneous system of linear equations (A - λᵢI)x = 0. This system represents the eigenvectors associated with λᵢ. The solutions to this system form the eigenspace corresponding to λᵢ.

3. Find the Solutions (Eigenvectors): Solving the system (A - λᵢI)x = 0 often involves techniques like Gaussian elimination or row reduction to find the solutions. These solutions are the eigenvectors associated with the eigenvalue λᵢ.

4. Determine a Basis: The set of linearly independent eigenvectors you found in step 3 forms a basis for the eigenspace corresponding to λᵢ. If you have multiple linearly independent eigenvectors, they constitute the basis; if there's only one, that single vector forms the basis. Linear independence can be checked using various methods, such as examining the vectors' linear combination or calculating the determinant of the matrix formed by the eigenvectors.

Example:

Let's consider the matrix A = [[2, 1], [1, 2]].

1. Find Eigenvalues: The characteristic equation is det(A - λI) = det([[2-λ, 1], [1, 2-λ]]) = (2-λ)² - 1 = 0. Solving this quadratic equation gives λ₁ = 1 and λ₂ = 3.

2. Solve (A - λI)x = 0 for each eigenvalue:

   • For λ₁ = 1: (A - I)x = [[1, 1], [1, 1]]x = 0. This leads to the equation x₁ + x₂ = 0, which means x₂ = -x₁. The eigenvectors are of the form x = [[x₁], [-x₁]] = x₁[[1], [-1]]. A basis for this eigenspace is { [[1], [-1]] }.

   • For λ₂ = 3: (A - 3I)x = [[-1, 1], [1, -1]]x = 0. This simplifies to x₁ = x₂. The eigenvectors are of the form x = [[x₁], [x₁]] = x₁[[1], [1]]. A basis for this eigenspace is { [[1], [1]] }.

Therefore, the eigenspace corresponding to λ₁ = 1 has a basis { [[1], [-1]] }, and the eigenspace corresponding to λ₂ = 3 has a basis { [[1], [1]] }.

Geometric Interpretation:

Eigenspaces represent subspaces that are invariant under the linear transformation represented by the matrix. This means that any vector within the eigenspace is only scaled (stretched or compressed) by the corresponding eigenvalue when the transformation is applied. The basis vectors of the eigenspace provide a way to describe this invariant subspace.

Conclusion:

Finding the basis for an eigenspace involves a systematic process of finding eigenvalues, solving a system of linear equations, and determining linearly independent eigenvectors. Understanding this process is fundamental to mastering various concepts within linear algebra and its applications. This method provides a powerful tool for analyzing the structure and behavior of linear transformations.