Proof Hint: Show +-1 are Eigenvalues of T on nxn Matrices with Real Entries

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In summary, an eigenvalue is a scalar value that represents how a linear transformation affects a given vector. To find eigenvalues, the characteristic equation of the matrix needs to be solved. In this proof, the +-1 values that make the determinant equal to zero are considered eigenvalues because they are the real roots of the characteristic equation. These eigenvalues have a specific effect on the given vector, either leaving it unchanged or reflecting it along its axis. This proof is useful in demonstrating the concept and significance of eigenvalues in linear algebra, and it has practical applications in various fields of mathematics.
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Hi everyone,

Let T be a linear operator on nxn Matrices with real entries defined by
T(A) = transpose(A).

Show that +-1 are the only eigenvalues of T.


Any tips on how to start this. I thought about writing the matrix representation relative to the standard basis, but it seemed really messy/tedious to write that out in general. Is there an easier way, or is that the only way to go?
 
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use the fact that T^2 = id
 
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To start, we can write out the matrix representation of T with respect to the standard basis. This will give us a better understanding of the linear transformation and allow us to see how the eigenvalues behave. Let's consider a generic nxn matrix A = (a_ij) with real entries. Then, T(A) = (b_ij) where b_ij = a_ji (the transpose of A).

Now, let's consider the eigenvector equation T(A) = cA where c is an eigenvalue and A is an eigenvector. Substituting in our expression for T(A), we get (b_ij) = c(a_ij). This means that for each entry in the matrix, we have b_ij = ca_ij.

We can see that this is only possible if c = 1 or c = -1. In other words, the only possible eigenvalues for T are +1 and -1. To prove that these are indeed eigenvalues, we can choose specific matrices A and see what happens when we apply T to them.

For example, let's take A = (1 0; 0 1). Applying T, we get T(A) = (1 0; 0 1) = A. This means that A is an eigenvector with eigenvalue +1. Similarly, if we take A = (0 1; 1 0), we get T(A) = (0 1; 1 0) = -A. This shows that A is an eigenvector with eigenvalue -1.

Thus, we can conclude that the only possible eigenvalues for T are +1 and -1. This is because the linear transformation T(A) = transpose(A) essentially flips the matrix about its main diagonal, which corresponds to multiplying by +1 or -1. Therefore, +1 and -1 are the only eigenvalues of T on nxn matrices with real entries.
 

1. What is an eigenvalue?

An eigenvalue is a scalar value that represents how a linear transformation affects a given vector. In other words, it is the value by which the vector is scaled after the transformation.

2. How do you find eigenvalues?

To find eigenvalues, you need to solve the characteristic equation of the matrix. This is done by subtracting the scalar value from the diagonal elements of the matrix and finding the determinant. The values that make the determinant equal to zero are the eigenvalues.

3. Why are +-1 considered eigenvalues in this proof?

In this proof, the matrix being examined is a nxn matrix with real entries. When a nxn matrix has real entries, its characteristic equation will always have at least one real root. Since the characteristic equation is solved to find eigenvalues, the real roots of the characteristic equation will also be the eigenvalues. Therefore, in this specific case, the +-1 that make the determinant equal to zero are the eigenvalues.

4. What does it mean if +-1 are eigenvalues of T on nxn matrices with real entries?

If +-1 are eigenvalues of T on nxn matrices with real entries, it means that the linear transformation T has a specific effect on the given vector. In this case, the vector is either left unchanged or is reflected along its axis depending on whether the eigenvalue is 1 or -1, respectively.

5. How is this proof useful in the field of mathematics?

This proof demonstrates the concept of eigenvalues and their significance in linear algebra. It also shows how specific values can be identified as eigenvalues for a given matrix. This knowledge is widely used in various fields of mathematics, including differential equations, quantum mechanics, and computer science.

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