Suppose T in L(v). Prove that 9 is an eigenvalue if T^2 if and only if -3 or 3 is an eigenvalue if T

Let's break down the proof into two parts: proving the forward direction (i.e., if 9 is an eigenvalue of \( T^2 \), then 3 or \(-3\) is an eigenvalue of \( T \)), and the reverse direction (i.e., if 3 or \(-3\) is an eigenvalue of \( T \), then 9 is an eigenvalue of \( T^2 \)).

### Forward Direction: 
**(If \( 9 \) is an eigenvalue of \( T^2 \), then 3 or \(-3\) is an eigenvalue of \( T \)).**

Assume \( 9 \) is an eigenvalue of \( T^2 \). By definition, this means there exists a nonzero vector \( v \in V \) such that:
\[
T^2(v) = 9v
\]
We can rewrite this as:
\[
(T^2 - 9I)(v) = 0
\]
This shows that \( v \) is an eigenvector of \( T^2 \) with eigenvalue 9.

Now, consider the polynomial \( x^2 - 9 = (x - 3)(x + 3) \). Since \( T^2(v) = 9v \), it follows that:
\[
(T^2 - 9I)(v) = (T - 3I)(T + 3I)(v) = 0
\]
This implies \( (T - 3I)(T + 3I)(v) = 0 \).

To satisfy this equation, \( v \) must be in the null space of \( T - 3I \) or \( T + 3I \), meaning:
\[
(T - 3I)(v) = 0 \quad \text{or} \quad (T + 3I)(v) = 0
\]
Thus, \( v \) is an eigenvector of \( T \) with eigenvalue 3 or \(-3\).

### Reverse Direction: 
**(If 3 or \(-3\) is an eigenvalue of \( T \), then 9 is an eigenvalue of \( T^2 \)).**

Assume \( \lambda = 3 \) or \( \lambda = -3 \) is an eigenvalue of \( T \). This means there exists a nonzero vector \( v \in V \) such that:
\[
T(v) = \lambda v
\]
Squaring both sides of this equation gives:
\[
T^2(v) = T(T(v)) = T(\lambda v) = \lambda T(v) = \lambda^2 v
\]
Substituting \( \lambda = 3 \) or \( \lambda = -3 \) into \( \lambda^2 \) yields:
\[
T^2(v) = 9v
\]
This shows that \( v \) is an eigenvector of \( T^2 \) with eigenvalue 9.

### Conclusion:
We've shown that \( 9 \) is an eigenvalue of \( T^2 \) if and only if 3 or \(-3\) is an eigenvalue of \( T \).

Would you like further details on any step, or have any questions?

Here are five related questions to explore this topic further:

1. What is the relationship between the eigenvalues of \( T^k \) and \( T \) in general?
2. How do the eigenvectors of \( T \) relate to those of \( T^2 \) and \( T^k \)?
3. What happens if 9 is a repeated eigenvalue of \( T^2 \)? What can be said about \( T \)?
4. Can the concept be extended to non-square matrices or infinite-dimensional spaces?
5. How would the proof change if we consider complex eigenvalues instead of real ones?

**Tip:** Understanding the behavior of polynomials of operators (like \( T^2 \), \( T^3 \), etc.) helps in analyzing the structure and properties of linear transformations more deeply.

Explore the mathematical proof that 9 is an eigenvalue of T^2 if and only if 3 or -3 is an eigenvalue of T. This detailed analysis covers linear algebra concepts such as eigenvalues, eigenfunctions, and polynomials of operators, relevant theorems, and their applications.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation