If is a matrix, and satisfy .
vice verse, If matrix satisfy the condition : ,
then not always Idempotent matrix, such as: (corrected)
can we add some conditions to such that is a Idempotent matrix
And since from the condition it follows that the minimal pol. of divides , we get what we want (if the matrix has rank 2 then zero can't be one of its eigenvectors thus its characteristic pol. is
For matrices of order greater than 2 I'm not sure but I doubt it.
Now let be a basis for ran(A) (so that k = rank(A)), and let be a basis for ker(A). Then is a basis for V, and the matrix of the linear transformation A with respect to this basis is a diagonal matrix with k 1s in the first k places on the diagonal, and 0s in the remaining places on the diagonal. This matrix obviously has trace k. But it is similar to the original matrix A, and so has the same trace as A. Therefore trace(A) = k = rank(A).