Thread: Eigenvalues 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.

Originally Posted by somestudent2

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) You should think about and . Are they always the same?

b) Shouldn't be too hard to play around with a couple of 2x2 matrices, see what happens with them ......

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

Originally Posted by somestudent2

would that be sufficient enough to say that since det(A) = det(A^t) Mr F says: No!

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

..

that is not what I meant mr F, I wrote that det(A)=det(A^t).....

Originally Posted by somestudent2

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

"then det(A-xI)=det(A^t-xI) since we pretty much subtract the same number .." is waaaay too vague ... you need to show, using a mathematical argument, how det(A) = det(A^t) leads to det(A-xI)=det(A^t-xI) ......

Originally Posted by somestudent2

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: .

Note 2: using .

But .

Therefore .

Therefore ......