a)Show that for any square matrix A, A^t (A transpose) and A have the same characteristic polynomial and hence the same eigenvalues.
b)Give an example of a 2x2 matrix A for which A^t and A have different eigenspaces.
Thank you.
a)Show that for any square matrix A, A^t (A transpose) and A have the same characteristic polynomial and hence the same eigenvalues.
b)Give an example of a 2x2 matrix A for which A^t and A have different eigenspaces.
Thank you.
would that be sufficient enough to say that since det(A) = det(A^t)
then det(A-xI)=det(A^t-xI) since we pretty much subtract the same number, I also considered this solution except it seemed too short to me to be true?
thanks again
Note 1: $\displaystyle (A - \lambda I)^T = A^T - \lambda I^T = A^T - \lambda I$.
Note 2: $\displaystyle \text{det} (A - \lambda I) = \text{det} (A - \lambda I)^T$ using $\displaystyle \text{det} B = \text{det} B^T$.
But $\displaystyle (A - \lambda I)^T = A^T - \lambda I$.
Therefore $\displaystyle \text{det} (A - \lambda I)^T = \text{det} (A^T - \lambda I)$.
Therefore ......