Let T:Mn(R)→Mn(R)
 be defined by T(A)=At
, the transpose of A. Then the sum of the distinct eigen values of T is
(Enter 1000 of there are no eigen values)

To solve this problem, we need to find the distinct eigenvalues of the linear transformation \( T: M_n(\mathbb{R}) \to M_n(\mathbb{R}) \) defined by \( T(A) = A^T \), where \( A^T \) is the transpose of the matrix \( A \).

### Step-by-step Solution:

1. **Understanding the Transformation:**
   - \( T \) takes an \( n \times n \) matrix \( A \) and maps it to its transpose \( A^T \).
   - The space \( M_n(\mathbb{R}) \) consists of all \( n \times n \) real matrices, which is a vector space of dimension \( n^2 \).

2. **Properties of Transpose:**
   - The transpose operation has some key properties:
     - \( (A^T)^T = A \)
     - \( (A + B)^T = A^T + B^T \)
     - \( (cA)^T = cA^T \) for any scalar \( c \)

3. **Possible Eigenvalues of the Transpose Map \( T \):**
   - Consider the effect of \( T \) on a matrix:
     - If \( A = A^T \) (i.e., \( A \) is symmetric), then \( T(A) = A \). In this case, \( A \) is an eigenvector of \( T \) with eigenvalue \( 1 \).
     - If \( A = -A^T \) (i.e., \( A \) is skew-symmetric), then \( T(A) = -A \). In this case, \( A \) is an eigenvector of \( T \) with eigenvalue \( -1 \).

4. **Finding Distinct Eigenvalues:**
   - The only two possibilities for eigenvalues of the transpose operator \( T \) are \( 1 \) and \( -1 \).

5. **Sum of Distinct Eigenvalues:**
   - The distinct eigenvalues of \( T \) are \( 1 \) and \( -1 \).
   - The sum of these distinct eigenvalues is:
   \[
   1 + (-1) = 0.
   \]

### Conclusion:

The sum of the distinct eigenvalues of the linear transformation \( T \) defined by \( T(A) = A^T \) is **0**.

### Additional Questions to Explore:

1. What are the dimensions of the eigenspaces corresponding to the eigenvalues 1 and -1 for the transpose map?
2. Can the transpose operation be defined for non-square matrices? If so, what changes?
3. What is the characteristic polynomial of the transformation \( T \)?
4. How does the result change if \( T \) is defined on complex matrices \( M_n(\mathbb{C}) \) instead of real matrices \( M_n(\mathbb{R}) \)?
5. What are some applications of the transpose operation in linear algebra and other fields?

### Tip:

When analyzing linear transformations, always check the properties of the transformation and consider how it acts on different types of vectors or matrices to determine possible eigenvalues and eigenvectors.

Explore the sum of distinct eigenvalues of the linear transformation defined by T(A) = A^T. Understand the properties of matrix transpose, eigenvalues in linear algebra, and their implications. Suitable for advanced undergraduate students and enthusiasts of linear algebra.

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