How To Find Basis Of Eigenspace

How to Find the Basis of an Eigenspace

Finding the basis of an eigenspace is a crucial step in linear algebra, particularly when dealing with eigenvalues and eigenvectors. This process allows us to understand the structure and properties of a linear transformation. This article will guide you through the process, explaining the concepts and providing a step-by-step approach with examples. Understanding this process is key to mastering topics like diagonalization and solving systems of differential equations.

What is an Eigenspace?

Before diving into finding the basis, let's clarify what an eigenspace is. Given a square matrix A, an eigenspace corresponding to an eigenvalue λ is the set of all eigenvectors associated with that eigenvalue, along with the zero vector. In simpler terms, it's the subspace spanned by all the eigenvectors associated with a specific eigenvalue. Each eigenspace is a vector space itself.

Steps to Find the Basis of an Eigenspace:

1. Find the Eigenvalues: First, you need to determine the eigenvalues of the matrix A. This involves solving the characteristic equation, det(A - λI) = 0, where I is the identity matrix and λ represents the eigenvalues. This equation will yield a polynomial equation, and the roots of this polynomial are your eigenvalues.

2. For Each Eigenvalue, Find the Eigenvectors: For each eigenvalue λ you found in step 1, you need to find the corresponding eigenvectors. This involves solving the system of linear equations (A - λI)v = 0, where v is the eigenvector. This system will always have at least one solution (the trivial solution v = 0), but you're interested in the non-trivial solutions, which are the eigenvectors.

3. Solve the System of Equations: The solution to (A - λI)v = 0 is found using techniques like Gaussian elimination or row reduction to find the reduced row echelon form (RREF) of the augmented matrix [(A - λI) | 0]. The free variables in the solution represent the parameters that define the eigenvectors.

4. Express Eigenvectors in Parametric Form: The solutions to the system of equations will often be expressed in parametric form. This means the eigenvectors will be written as linear combinations of vectors, with the parameters as coefficients.

5. Identify the Basis Vectors: The linearly independent vectors in the parametric representation of the eigenvectors form the basis for the eigenspace corresponding to the eigenvalue λ. The number of linearly independent vectors equals the dimension of the eigenspace, also known as the geometric multiplicity of the eigenvalue.

Example:

Let's consider the matrix:

A = [[2, 1], [1, 2]]

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

2. Eigenvectors for λ₁ = 1: (A - I)v = 0 becomes:

   [[1, 1], [1, 1]]v = 0

   Row reducing leads to [[1, 1], [0, 0]]v = 0. The solution is v = t[-1, 1]ᵀ, where t is a parameter. Thus, a basis for the eigenspace corresponding to λ₁ = 1 is {[-1, 1]ᵀ}.

3. Eigenvectors for λ₂ = 3: (A - 3I)v = 0 becomes:

   [[-1, 1], [1, -1]]v = 0

   Row reducing leads to [[-1, 1], [0, 0]]v = 0. The solution is v = t[1, 1]ᵀ. Thus, a basis for the eigenspace corresponding to λ₂ = 3 is {[1, 1]ᵀ}.

Conclusion:

Finding the basis of an eigenspace is a fundamental skill in linear algebra. By following these steps and understanding the underlying concepts of eigenvalues, eigenvectors, and linear independence, you can effectively determine the basis for any eigenspace. Remember that the dimension of the eigenspace (the number of vectors in the basis) is always less than or equal to the algebraic multiplicity of the eigenvalue (the multiplicity of the eigenvalue as a root of the characteristic polynomial). This process is essential for understanding the properties of linear transformations and solving various problems in advanced mathematical applications.