Register to reply 
Coefficients of characteristic polynomial (linear algebra) 
Share this thread: 
#1
Feb610, 02:42 PM

P: 146

1. The problem statement, all variables and given/known data
A is an nxn square, real matrix. Let f(x) be the characteristic polynomial, write f(x) = x^{n}  c_{1}x^{n1} + ... + (1)^{r}c_{r}x^{nr} + ... + (1)^{n}c_{n} Show that c_{n1} = [tex]\sum[/tex] det (A_{ii}) where A_{ii} is the (i,i) minor of A. Similarly, what is the coefficient c_{r}? 2. Relevant equations 3. The attempt at a solution I have shown that c_{1} = trace(A) and c_{n} = det(A). c_{n1} is the coefficient of x, so is the sum of all products involving one entry from the diagonal, would this product then be the determinant of the matrix formed by deleting the row and column that this entry is in, so A_{ii}? If this is true, how would I express it more rigorously? Also I'm not sure how to generalise for c_{r}. 


#2
Feb610, 05:37 PM

P: 2,158

Write det(A x I) in the "sum over permutations" form. Then consider the form of the permuations that will get you a factor x from some specific position on the diagonal. You then consider that restricted sum over permutations and see if it corresponds to an unrestricted sum over permutations of a smaller set of rows and is thus, by definition, the determinant of a minor of the matrix.



Register to reply 
Related Discussions  
Linear Algebra  Proof of the characteristic polynomial for the inverse of A  Calculus & Beyond Homework  1  
Linear Algebra Minimal Polynomial  Calculus & Beyond Homework  0  
Characteristic polynomial splits into linear factors  Linear & Abstract Algebra  1  
Characteristic polynomial  Linear & Abstract Algebra  3  
Characteristic Polynomial  Calculus & Beyond Homework  11 