To show that $\|A + I\| > 1$ for a nilpotent matrix $A$, we'll use properties of nilpotent matrices and matrix norms. Here’s a detailed proof:

Definitions and Properties

1. Nilpotent Matrix: A matrix $A$ is nilpotent if $A^k = 0$ for some positive integer $k$.
2. Matrix Norm: A norm $\|\cdot\|$ is a function that assigns a non-negative length or size to a matrix. Common norms include the spectral norm, Frobenius norm, and $L_p$ norms. For this proof, we'll use a norm that is sub-multiplicative and satisfies $\|I\| = 1$.

Proof Steps

1. Nilpotency of $A$: Since $A$ is nilpotent, there exists an integer $m > 0$ such that $A^m = 0$ and $A^{m-1} \neq 0$.

2. Matrix Exponential Series: Consider the series expansion of $(A + I)^m$: $(A + I)^m = \sum_{k=0}^{m} \binom{m}{k} A^k I^{m-k}$ Given $A^m = 0$, the series simplifies to: $(A + I)^m = I$

3. Norm Properties: Using the sub-multiplicative property of the norm, we get: $\|(A + I)^m\| = \|I\| = 1$ If $\|A + I\| \leq 1$, then $\|(A + I)^m\| \leq 1$.

4. Contradiction: Assume $\|A + I\| \leq 1$. Then: $\|(A + I)^m\| \leq \|A + I\|^m \leq 1^m = 1$ However, since $(A + I)^m = I$, we know: $\|I\| = 1$ For $\|(A + I)^m\| = 1$, $\|A + I\|$ must be strictly greater than 1 to ensure that the product norm is exactly 1 after $m$ multiplications. This is because if $\|A + I\| \leq 1$, the repeated multiplication would result in a norm strictly less than 1 unless $\|A + I\| = 1$.

5. Conclusion: Therefore, the assumption $\|A + I\| \leq 1$ leads to a contradiction. Hence: $\|A + I\| > 1$

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Relative Questions:

1. What is the significance of the index of nilpotency in the context of matrix norms?
2. How can different types of matrix norms affect the properties of a nilpotent matrix?
3. What are some common applications of nilpotent matrices in mathematics and engineering?
4. Can you provide examples of nilpotent matrices in different dimensions?
5. How does the spectral norm of a matrix differ from other norms in terms of computational complexity?
6. What role does the identity matrix play in the context of matrix norms?
7. How can you prove the nilpotency of a given matrix using its eigenvalues?
8. What are some advanced properties of nilpotent matrices that are useful in linear algebra?

Tip:

Understanding the properties of matrix norms and how they interact with matrix operations is crucial for solving problems involving matrix inequalities and spectral analysis.