Proving the Invertibility of I A When A is a Nilpotent Matrix

Understanding the properties of nilpotent matrices and their implications on the invertibility of matrices in the form I A is crucial for advanced linear algebra and matrix theory. We will explore a detailed proof to show that if A is a nilpotent square matrix, the matrix I A is indeed invertible. This proof will be supplemented with alternative insights and a more general statement regarding eigenvalues of nilpotent matrices.

1. Definition and Preliminaries

A matrix A is called nilpotent if there exists a positive integer k, such that Ak 0, where 0 is the zero matrix.

2. Proof: Using Neumann Series

To prove that I A is invertible, we need to find a matrix B such that (I A)B I.

2.1 Nilpotency of A

Let A be a nilpotent matrix of order k, meaning that Ak 0 for some integer k.

2.2 Finding the Inverse Using Neumann Series

We propose that the inverse can be expressed in the form:

B I - A A2 - A3 ... (-1)k-1 Ak-1

This series is finite and truncates at Ak-1 because Ak 0.

2.3 Verification: (I A)B I

We need to check if (I A)B I:

(I A)(I - A A2 - A3 ... (-1)k-1 Ak-1)

Expanding the product, we get:

I - A A2 - A3 ... (-1)k-1 Ak-1 A - A2 A3 - ... (-1)k-1 Ak

Since Ak 0, the last term vanishes:

I (-1)k-1 (Ak-1 - Ak-1) I

3. Alternative Approach: Using Eigenvalues

An alternative and simpler approach involves eigenvalues. Suppose IA is invertible. This means it has no null vector. If u is a null vector such that Au -u, then:

Anu -An-1u ≠ 0 for the smallest n such that Anu 0, which is a contradiction because Anu ≠ 0 if n is the smallest integer making An 0.

4. Equivalence of Matrices

It is worth noting that:

(I - A A2 - A3 ... (-1)k-1 Ak-1) I - (-1;k-1Ak)

Since Ak 0, it simplifies to:

IA I - 0 I

5. Generalization: Eigenvalues of Nilpotent Matrices

A more interesting consequence is that if A λI is invertible for some λ ≠ 0, then A cannot have any eigenvalues other than 0. This leads us to the conclusion that all eigenvalues of a nilpotent matrix are 0, which is a much stronger and cleaner result than the initial problem statement.

Conclusion

We have rigorously demonstrated that if A is a nilpotent matrix, then I A is invertible. This result leverages both the definition of nilpotent matrices and the Neumann series, and it naturally extends to a more general statement about the eigenvalues of nilpotent matrices being exclusively 0. This insight provides a deeper understanding of the structure of nilpotent matrices and their implications in linear algebra.