Proving detA = λ1...λn for Real Eigenvalues

In summary, when given a nxn matrix A with real eigenvalues λ1,...,λn, repeated according to their multiplicities, we can prove that detA = λ1...λn by using the definitions of eigenvectors and the relationship between similar matrices and their determinants. Another approach is to consider the characteristic polynomial and use the fundamental theorem of algebra to factorize it.
  • #1
dsyh
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0

Homework Statement


Let A be nxn matrix, suppose n has real eigenvalues,λ1,...,λn, repeated according to multipilicities. Prove that detA = λ1...λn.

Homework Equations




The Attempt at a Solution


I started by applying the definition, Av = λv, where v is an eigenvector. then I just dun know how to keep going.. is there anyone can help me out? or at least give me some hints..
thx..
 
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  • #2
If matrices A and B are similar, what can you say about their determinants? Use this relationship to help you solve the problem.
 
  • #3
DH offered an correct approach,but you need to know something about similarity and Jordan normal form or Schur's lemma
Here's another approach: consider the characteristic polynomial det(A-λI),by fundamental theorem of algebra, it can be factorized into (λ1-λ)(λ2-λ)...(λn-λ),then let λ=0 and see what will happen
 

FAQ: Proving detA = λ1...λn for Real Eigenvalues

What is the definition of a real eigenvalue?

A real eigenvalue is a scalar that, when multiplied by a vector, gives back the same vector with only a change in magnitude (not direction).

Why is it important to prove that detA = λ1...λn for real eigenvalues?

Proving that detA = λ1...λn for real eigenvalues is important because it helps us understand the behavior of a matrix and its corresponding eigenvectors and eigenvalues. It also allows us to solve various problems in linear algebra and other fields of mathematics.

What are the steps involved in proving detA = λ1...λn for real eigenvalues?

The steps involved in proving detA = λ1...λn for real eigenvalues include finding the characteristic polynomial of the matrix, finding the roots of the polynomial (which will be the eigenvalues), and then using these eigenvalues to find the eigenvectors. Finally, the determinant of the matrix can be calculated using the product of the eigenvalues.

What is the significance of real eigenvalues in linear algebra?

Real eigenvalues play a crucial role in linear algebra as they provide information about the behavior and properties of a matrix. They are used to solve systems of linear equations, diagonalize matrices, and analyze the stability of dynamical systems.

Can a matrix have complex eigenvalues and still satisfy detA = λ1...λn for real eigenvalues?

No, if a matrix has complex eigenvalues, it cannot satisfy the equation detA = λ1...λn for real eigenvalues. The equation only holds for matrices with real eigenvalues.

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