A nilpotent show norm(A+I) greater than 1

To show that \(\|A + I\| > 1\) for a nilpotent matrix \(A\), let's proceed with the proof step by step.

### Definition and Properties
1. **Nilpotent Matrix**: A square matrix \(A\) is called nilpotent if there exists some positive integer \(k\) such that \(A^k = 0\). The smallest such \(k\) is called the index of nilpotency.

2. **Norm of a Matrix**: There are different types of norms for matrices, but commonly used norms include the spectral norm (induced by the Euclidean norm on \(\mathbb{R}^n\)), the Frobenius norm, and others. For the sake of this proof, we will use a norm that is sub-multiplicative and satisfies \(\|I\| = 1\).

### Proof Outline
1. **Nilpotency of \(A\)**: Since \(A\) is nilpotent, there exists an integer \(m >

Learn how to prove that the norm of the matrix A + I is greater than 1 for a nilpotent matrix A. This problem involves concepts from matrix theory and norms of matrices. Suitable for advanced undergraduate students and researchers in linear algebra.

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