1. ## characteristic polynomial question

If M is an n x n matrix with entries from a field F, then is there always a positive integer k such that M^k = summation(i=0, k-1) lamda_i*M^i.

If M=I_n is the n x n identity matrix, then what is the smallest k with this property?

2. hmmm its been a while since I worked with matrices (MATLAB to the rescue), but at first sight it seems like an eigenvalue problem. Im not sure if its possible to diaganolize the matrix, Im sorry its been a while:S.

take M = PDP^-1.
It should be possible to do this since you are dealing with a square matrix.

If you have done the diagonalization correctly you simple raise your lambda values to what ever k you wish to get M^k. This approach should be a start, if you havent covered matrix diagonalization by finding the eigenvectors of a matrix then Im not sure what other approach to take....but I dont think you need to do any diagonalization but simply consider the general theory of what is produced from it and consequently equate PDP^-1 with your summation.

3. Is there a connection between the multiplicity k, of the lamda's, and the matrix M^k

4. Originally Posted by Luck of the Irish
Is there a connection between the multiplicity k, of the lamda's, and the matrix M^k
Well yes because the diagonalized matrix represents your original matrix and an efficient way of calculating the result of raising M to any k-th power. Again I'm not sure if my approach to the problem is too simple but I think its worth a try.

5. Sorry I realized i didnt quite address your question. You diagonalized matrix consists of PDP^-1. The matrix D consists of your eigenvalues of Matrix M (original matrix) with eigenvalues as your diagonal entries. Rasing the eigenvalues to any k-th power will ultiamtely also raise your matrix M to the k-th power.

6. If M=I_n is the n x n identity matrix, then what is the smallest k with this property? I think the smallest is k=1. that should solve the summation for the identity, however i still not sure about the first question:

If M is an n x n matrix with entries from a field F, then is there always a positive integer k such that M^k = summation(i=0, k-1) lamda_i*M^i.

any other advice? for the first part?

7. have you tried the approach i suggested???