1. ## Transponse and Eigenvalues

Let A be a Matrix with no real eigenvalues. It is possible that the following Matrix B has real eigenvalues?
B=A-A*AT+AT

If I multiply a Matrix with it's transponse I get a symmetric matrix, which has two real roots so A*AT has real eigenvalues. If I subtract A and add AT it is not symmetric anymore... but how can I derive if it still has no real eigenvalues?

2. ## Re: Transponse and Eigenvalues

Suppose that $\lambda, \underline{x}$ is an eigenpair of $A$ (and $A^{T}),$

then if

$B=A-AA^{T}+A^{T},$

$B\underline{x}=(A-AA^{T}+A^{T})\underline{x}=A\underline{x}-AA^{T}\underline{x}+A^{T}\underline{x}=\lambda \underline{x}-A\lambda\underline{x}+\lambda\underline{x}=(2 \lambda-\lambda^{2})\underline{x}.$

$\therefore B$ will only have real eigenvalues if $2 \lambda - \lambda^{2}$ is real and positive.

3. ## Re: Transponse and Eigenvalues

Ooops ! Sorry about that, $2 \lambda - \lambda^{2}$ needs to be real but not necessarily positive.

Also, it would appear that it is possible to come up with a condition for the eigenvalues of $B$ to be real.

4. ## Re: Transponse and Eigenvalues

Thank you, what would happen if A has no real eigenvalue. Is it possible that A+AT has a real eigenvalue?

5. ## Re: Transponse and Eigenvalues

Profuse and sincere apologies for those two responses, I suspect it was those magic mushrooms.

The idea of post multipling by $\underline{x}$ isn't correct because even though $A$ and $A^{T}$ have the same eigenvalues, they will (usually) have different eigenvectors.

The basic property that answers both of your questions is that the eigenvalues of a real symmetric matrix are real.

So, even if the matrix $A$ (with all real elements) has complex eigenvalues, the matrices $C=A+A^{T}$ and $D=AA^{T}$ will both have real eigenvalues since they are both symmetric.

For your earlier example $B=A-AA^{T}+A^{T},$ this is also symmetric, (consider it as $B=(A+A^{T})-AA^{T})$ and will also have real eigenvalues irrespective of whether the eigenvalues of $A$ are real or complex.