Register to reply 
Determinants and Eigenvalues 
Share this thread: 
#1
Oct2809, 10:09 AM

P: 29

1. The problem statement
For integers m >= n, Prove det(xI_{m}  AB) = x^{mn}det(xI_{n}  BA) for any x in R. 2. Relevant equations A is an m x n matrix B is an n x m matrix 3. The attempt at a solution I tried working out the characteristic polynomials by hand but it just seems too tedious for a nice proof. I know that each x is an eigenvalue of AB but after that I'm stumped. 


#2
Oct2809, 02:22 PM

P: 28

Sylvester's determinant theorem covers this: http://en.wikipedia.org/wiki/Determi...minant_theorem
A link to the proof is in the citation. 


Register to reply 
Related Discussions  
Similar matrices = Same Eigenvalues (NO DETERMINANTS!)  Calculus & Beyond Homework  12  
Eigenvalues and determinants  Calculus & Beyond Homework  9  
Definition of the determinant i = 1  Linear & Abstract Algebra  3  
More determinants!  Calculus & Beyond Homework  7  
Linear algebra: determinants and eigenvalues  Linear & Abstract Algebra  15 