Hi
This web site has been very helpful and I have one last problem with my eigenvalue chapter that I have been struggling with maybe someone can help. It is not a problem but the hint to the problem. I have been able to work the problem using the hint OK but am stuck on the hint. I can see the hint is true but cannot prove it in the general case. Here it is : Let lambda = L , lambda being the symbol for the eigenvalue representation in the characteristic polynomial and L1,L2,...,Ln all the specific eigenvalues. I = identity matirix
For an nXn genereal matrix how do I prove determinant( A - LI) = (-1)^n( L-L1)(L-L2),...(L-Ln) . The problem is with the (-1)^n in front.
I can prove det(LI-A) = (-1)^n(A-LI) but it does not seen to help. The factor theorem tells me I can factor the characteristic polynomial into factors of the form (L-Ln) but when n is odd there is ONE factor that is (Ln-L). How can I prove this hint with the factor theorem or any other theorem is a complete mystery to me. I can see it is always the case by working examples but it seems challenging to prove it. Can anyone please give a shot at this for me.
I would be very thankful
Julius
This web site has been very helpful and I have one last problem with my eigenvalue chapter that I have been struggling with maybe someone can help. It is not a problem but the hint to the problem. I have been able to work the problem using the hint OK but am stuck on the hint. I can see the hint is true but cannot prove it in the general case. Here it is : Let lambda = L , lambda being the symbol for the eigenvalue representation in the characteristic polynomial and L1,L2,...,Ln all the specific eigenvalues. I = identity matirix
For an nXn genereal matrix how do I prove determinant( A - LI) = (-1)^n( L-L1)(L-L2),...(L-Ln) . The problem is with the (-1)^n in front.
I can prove det(LI-A) = (-1)^n(A-LI) but it does not seen to help. The factor theorem tells me I can factor the characteristic polynomial into factors of the form (L-Ln) but when n is odd there is ONE factor that is (Ln-L). How can I prove this hint with the factor theorem or any other theorem is a complete mystery to me. I can see it is always the case by working examples but it seems challenging to prove it. Can anyone please give a shot at this for me.
I would be very thankful
Julius
