# Help Showing d/dt of Char. Polynomial

1. Feb 14, 2005

### oobob

I've been working on a problem all day but I can't seem to go anywhere with it. I have to prove the following equation holds, in which $$P_A(t)$$ refers to the characteristic polynomial of an n by n matrix A and is expressed in terms of t. Also, $$P_A_i(t)$$ is the characteristic polynomial of the principal submatrix formed by removing row i and column i from A, also expressed in terms of t. The equation is:

$$\frac{d}{dt}P_A(t) = \sum_{i=1}^n P_A_i(t)$$

I've been trying to prove it by induction, using the following formula for the left side. In the formula, for a given k, $$E_k(A)$$ is the sum of the k-by-k principal minors of A.:

$$\frac{d}{dt}P_A(t) = nt^n^-^1 - (n-1)E_1(A)t^n^-^2 + (n-2)E_2(A)t^n - ... \pm E_n_-_1(A)$$

I'm not sure how to arrive at the right side of the first equation at all. I've tried rearranging it, but I don't know how to make up for the fact that all of the (n-1) by (n-1) principal submatrices of A are different. My only interpretation of the right hand side is as the sum of the characteristic polynomials of all (n-1) by (n-1) principal submatrices of A, but I can't seem to draw anything from that. Am I on an entirely wrong path, or am I just missing a step somewhere?

Thanks,
Oobob

Last edited: Feb 14, 2005