Eigenvectors and Nullspace: A Closer Look

Understanding the relationship between eigenvectors and the nullspace of a matrix can be crucial for various applications in linear algebra and beyond. In this article, we will explore the conditions under which eigenvectors of a matrix A are in the nullspace of I - A, where I is the appropriately sized identity matrix.

Introduction to Eigenvectors and Nullspace

Eigenvectors and eigenvalues are fundamental concepts in linear algebra. An eigenvector of a matrix A is a non-zero vector v such that when A is multiplied by v, the result is a scalar multiple of v. This scalar is known as the eigenvalue, denoted by λ. Mathematically, this can be written as:

Av λv

The nullspace (or kernel) of a matrix is the set of all vectors v such that Av 0. In other words, these are the vectors that get mapped to the zero vector under the transformation defined by A.

Conditions for Eigenvectors in Nullspace of I - A

The statement that eigenvectors of matrix A are necessarily in the nullspace of I - A is not always true. However, there is a special case where this is true, and that is when the eigenvalue is 1.

Deriving the Condition

Let's see why only the eigenvectors associated with the eigenvalue 1 are in the nullspace of I - A. Consider the equation:

(I - A)v 0

This equation can be rewritten as:

Iv - Av 0

Simplifying, we get:

Av Iv

In a standard basis, Iv v, so:

Av v

This is the definition of an eigenvector with eigenvalue 1:

Av 1v

Therefore, for an eigenvector v to be in the nullspace of I - A, it must satisfy:

Av v

which implies that the eigenvalue λ must be 1. Thus, the eigenvectors of A that lie in the nullspace of I - A are precisely those corresponding to the eigenvalue 1.

Eigenvectors of A with Eigenvalue r ≠ 1

For eigenvectors corresponding to eigenvalues r ≠ 1, the nullspace of I - A does not contain them. To understand this, consider the general form of the eigenvector equation:

(rI - A)v 0

This equation can be rewritten as:

rIv - Av 0

Simplifying, we get:

Av rv

For r ≠ 1, solving for v in the equation (1 - rI A)v 0 does not yield the zero vector, indicating that the eigenvectors corresponding to r ≠ 1 are not in the nullspace of I - A.

Conclusion

In summary, while it is not true that all eigenvectors of a matrix A lie in the nullspace of I - A, it is true that the eigenvectors corresponding to the eigenvalue 1 do. This relationship is a special case and is often useful in various mathematical and engineering applications. Understanding these concepts deeply can provide valuable insights into the behavior of linear transformations and the properties of matrices.