• Support PF! Buy your school textbooks, materials and every day products Here!

Characteristic Eq for Matrix problem

  • Thread starter Zoe-b
  • Start date
  • #1
98
0

Homework Statement


Where A is an n x n matrix and I is the n x n identity
In the expansion of det(x*I-A), show that the coefficient of x is equal to the sum from i = 1 to n of the determinant of the Aii minor. (where Aii = the submatrix of A formed by deleting row i and column i)


Homework Equations


if X1, X2, ... Xn are the eigenvalues of A then the coefficient of x is equal to the sum from i = 1 to n of (product of eigenvalues)/Xi.


The Attempt at a Solution


I have tried quite a few different ways of doing this but getting nowhere..
using the definition of a determinant as a sum of products over permuations of Sym (n) and then using the Leibnix rule for integrals I seem to find that (where a11 is the top left entry of A and so on)
d(det(x*I-A)/dx = sum (from i =1 to n) of det(x*I-A)/(x-aii)

The constant coefficient of this will be equal to the x coefficient of det(x*I-A) but the RHS only cancels directly to what I want if aii is an eigenvalue, which obviously isn't always the case. I tried to use row permutations but this obviously changes the Aii minors and so doesn't seem to work. Any ideas? I was trying to follow the proof on

http://www.sciencedirect.com/scienc...26cc582e2a0f1358d2f4b36931f4954e&searchtype=a

(sorry for the long link!) but the notation is a bit beyond me. Am I on the right track at all or is there a better way to attempt this? Sorry for not including full working but without the maths language its near impossible to be clear.
Thanks
 

Answers and Replies

  • #2
hunt_mat
Homework Helper
1,739
18
Can you prove it for when A is diagonal
 
  • #3
98
0
Yes I understand for when it is diagonal but altho row/column operations of form (row i -> row i + d*row j) won't change the determinant of the matrix, they will change the determinant of the minors.. I don't see how to relate the diagonal form back to the original matrix.
 
  • #4
hunt_mat
Homework Helper
1,739
18
How about for a diagonaliable matrix? That is the key here.
 
  • #5
98
0
Hmmmn I'm still pretty clueless but I'll give it another go tomorrow! Thanks
 

Related Threads on Characteristic Eq for Matrix problem

  • Last Post
Replies
6
Views
4K
  • Last Post
Replies
2
Views
1K
Replies
5
Views
8K
Replies
2
Views
985
  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
1
Views
906
Replies
0
Views
2K
Replies
2
Views
2K
Replies
5
Views
548
Top