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.(Whew)

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- Jan 22nd 2008, 01:26 PMsomestudent2Eigenvalues and Eigenvectors, PLEASE
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.(Whew) - Jan 22nd 2008, 05:34 PMmr fantastic
- Jan 22nd 2008, 06:07 PMsomestudent2
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 - Jan 22nd 2008, 08:22 PMmr fantastic
- Jan 22nd 2008, 09:15 PMsomestudent2
that is not what I meant mr F, I wrote that det(A)=det(A^t).....

- Jan 22nd 2008, 10:38 PMmr fantastic
- Jan 23rd 2008, 01:27 AMmr fantastic
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 ......