# Eigenvectors and Values for T(A)=A^t

#### kaelbu

I'm having some trouble getting started with this, if someone could point me in the right direction that would awesome.
If T(A) = A^t (A-transpose) show that +1 and - 1 are the only eigenvalues of T.
I think that my problem is that I don't know how to symbolize the matrix of linear transformation for T, but perhaps there is another way to go about doing this problem.
Thanks!

MHF Helper
Accidental post

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#### HallsofIvy

MHF Helper
T is a linear operation on matrices that maps a matrix into its transpose? I started to say "Think about the fact that $$\displaystyle A$$ and $$\displaystyle A^T$$ have the same main diagonal", but then it occured to me that A might not be a square matrix. Certainly, you can look at $$\displaystyle a_{11}$$ and $$\displaystyle a_22$$ which must be the same in both matrices.

#### Opalg

MHF Hall of Honor
I'm having some trouble getting started with this, if someone could point me in the right direction that would awesome.
If T(A) = A^t (A-transpose) show that +1 and - 1 are the only eigenvalues of T.
I think that my problem is that I don't know how to symbolize the matrix of linear transformation for T, but perhaps there is another way to go about doing this problem.
Hint: $$\displaystyle T^2$$ is the identity.

• HallsofIvy

#### kaelbu

I started to say "Think about the fact that $$\displaystyle A$$ and $$\displaystyle A^T$$ have the same main diagonal", but then it occured to me that A might not be a square matrix.
Isn't it true that for a matrix to have transpose it must be square?
Also, I the fact that A is a square matrix was actually part of the problem (T is a linear operator on $$\displaystyle M_(nxn)(F)$$).
Sorry I did not include that in the original problem, I guess I have a bad habit of over looking that part of the problem because I don't really have any idea what's going on, therefore it doesn't really make any difference to me what vector space the transformations in.

#### kaelbu

Oh!
If T^2 is the identity, T must also be the identity, I think. Oh no, each diagonal entry must be the $$\displaystyle \sqrt(1) = + or - 1$$.
Thanks! You're the greatest!

#### Opalg

MHF Hall of Honor
Oh!
If T^2 is the identity, T must also be the identity, I think. Oh no, each diagonal entry must be the $$\displaystyle \sqrt(1) = + or - 1$$.
Thanks! You're the greatest!
Thanks for the compliment, but I'm not sure what you mean by the diagonal entries of T?

T is defined by $$\displaystyle T(A) = A^{\textsc{t}}$$. So $$\displaystyle T^2(A)= (A^{\textsc{t}})^{\textsc{t}} = A$$. And $$\displaystyle \lambda$$ is an eigenvalue of T if there is a nonzero matrix A such that $$\displaystyle T(A) = \lambda A$$. Then $$\displaystyle A = T^2(A) = T(T(A)) = T(\lambda A) = \lambda T(A) = \lambda^2A$$. So $$\displaystyle (\lambda^2-1)A = 0$$ and hence $$\displaystyle \lambda^2=1$$.

#### kaelbu

That's more or less what I meant, though I did it slightly differently.
Thanks again for the help.