STEM Problem Detail

Problem Statement

Prompt

Provide me stepwise solution of question 1.25.

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## Problem 1.25

Let $$A \in \mathbb{C}^{n \times n}$$ ($$n > 1$$) be nilpotent of index $$n$$ (that is, $$A^n = 0$$ but $$A^{n-1} \neq 0$$).

**Part 1:** Show that $$A$$ has no square root.

**Part 2:** Show that $$A + cA^{n-1}$$ is a square root of $$A^2$$ for any $$c \in \mathbb{C}$$.

**Part 3:** Describe all such $$A$$.

**Solution:**

1. **Properties of $$A$$:**

- $$A^n = 0$$, $$A^{n-1} \neq 0$$
- All eigenvalues of $$A$$ are 0
- Minimal polynomial: $$\psi(t) = t^n$$
- Characteristic polynomial: $$p(t) = t^n$$
2. **Part 1: Show $$A$$ has no square root.**

**Method 1 (Using Jordan form):**

- Since $$A^{n-1} \neq 0$$ but $$A^n = 0$$, the Jordan form of $$A$$ must contain a single $$n \times n$$ Jordan block $$J_n(0)$$.
- By Theorem 1.22, $$J_n(0)$$ has no square root because $$f(t) = t^{1/2}$$ is not defined on the spectrum of $$J_n(0)$$ as a primary matrix function.
- Since $$A$$ is similar to $$J_n(0)$$, $$A$$ has no square root.

**Method 2 (Elementary proof):**

- Suppose $$A = X^2$$ for some $$X \in \mathbb{C}^{n \times n}$$.
- Then $$X^{2n} = A^n = 0$$, so $$X$$ is nilpotent.
- Since $$X$$ is $$n \times n$$, $$X^n = 0$$.
- Then $$A^{n-1} = X^{2(n-1)} = X^n X^{n-2} = 0 \cdot X^{n-2} = 0$$.
- This contradicts $$A^{n-1} \neq 0$$.
- Therefore, no such $$X$$ exists.
3. **Part 2: Show $$(A + cA^{n-1})^2 = A^2$$ for any $$c \in \mathbb{C}$$.**

Compute:
\[
(A + cA^{n-1})^2 = A^2 + cA A^{n-1} + cA^{n-1}A + c^2 A^{n-1}A^{n-1}
\]
\[
= A^2 + cA^n + cA^n + c^2 A^{2n-2}
\]

Now analyze each term:

- $$A^n = 0$$ by nilpotency
- $$A^{2n-2} = A^n A^{n-2} = 0 \cdot A^{n-2} = 0$$ since $$n > 1$$

Therefore:
\[
(A + cA^{n-1})^2 = A^2
\]

So $$A + cA^{n-1}$$ is a square root of $$A^2$$ for any $$c \in \mathbb{C}$$.
4. **Part 3: Describe all such $$A$$.**

- $$A$$ is similar to the $$n \times n$$ Jordan block with eigenvalue 0:
\[
J_n(0) = \begin{bmatrix}
0 & 1 & 0 & \cdots & 0 \\
0 & 0 & 1 & \cdots & 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
0 & 0 & 0 & \cdots & 1 \\
0 & 0 & 0 & \cdots & 0
\end{bmatrix}
\]
- All such matrices $$A$$ are of the form:
\[
A = P J_n(0) P^{-1}
\]
for some invertible $$P \in \mathbb{C}^{n \times n}$$.
- Alternatively, $$A$$ can be characterized by:

1. **Additional observation:** The family $$A + cA^{n-1}$$ gives infinitely many square roots of $$A^2$$, showing that square roots are not unique even when they exist.

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Problem #7