Eigenvalues and determinants

In summary, the eigenvalues of an nxn matrix are the roots of the characteristic polynomial p(lambda). The multiplicity of the eigenvalue is the multiplicity of the root. If lambda=0 on both sides, then det(A) =(lambda_1)(lambda_2)...(lambda_n).
  • #1

Homework Statement


Let A be an nxn matrix, and suppose A has n real eigenvalues lambda_1, ...lambda_n repeated according to multiplicities. Prove that det A = lambda_1...lambda_n


Homework Equations


None


The Attempt at a Solution


Could someone explain what is meant by 'repeated according to multiplicities'? and give me a hint as to how to start this proof? thank u ~~
 
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  • #2
The eigenvalues are the roots of the characteristic polynomial p(lambda)=det(I*lambda-A), the multiplicity of the eigenvalue is the multiplicity of the root. E.g. (lambda-1)^2 has a root 1 of multiplicity 2. Roots of polynomials correspond to linear factors of the polynomial. Think about how to find p(0).
 
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  • #3
to find p(0), i wud just sub 0 in the place of every variable and solve. I will be left with a constant, if there is a one
Is this wat u r asking? but how does this relate to the question? =S
 
  • #4
I'm asking you to write out the characteristic polynomial in terms of lambda and lambda_1...lambda_n. Then think about what constant you get if lambda=0. How is it related to lambda_1...lambda_n? And how is that constant related to det(A)?
 
  • #5
How can i write our the characteristic polynomial if we're dealing with a general nxn matrix?
 
  • #6
You know the eigenvalues. E.g. if lambda_1 is a eigenvalue, then it's a root of the characteristic polynomial p(lambda). That means p(lambda) has a factor (lambda-lambda_1), right? Remember what you know about how roots of polynomials are related to factors of polynomials.
 
  • #7
the factors of polynomials are the roots of the polynomials

i think...
det(A)=(lambda-lambda_1)(lambda-lambda_2)...(lambda-lambda_n) so the eigenvalues are lambda_1...lambda_n
 
  • #8
Ok, but that's not quite det(A), it's det(lambda*I-A). det(A) doesn't have a lambda in it. So if lambda=0 on both sides what do you get?
 
  • #9
ohhhhhh i think i got it now

det(A-lambda*I) =(lambda-lambda_1)(lambda-lambda_2)...(lambda-lambda_n)
if lambda =0, then we have
det(A) =(lambda_1)(lambda_2)...(lambda_n)

but, can we just set lambda = 0 like that?
 
  • #10
Yes, you can. But you have to pay attention to the minus signs. And the characteristic polynomial is det(lambda*I-A). det(A) and det(-A) are different. But that's pretty close.
 

1. What are eigenvalues and eigenvectors?

Eigenvalues and eigenvectors are concepts in linear algebra that are used to understand the behavior of a linear transformation. Eigenvalues are the scalar values that represent how the transformation changes the magnitude of the eigenvectors, which are the vectors that remain in the same direction after the transformation.

2. What is the significance of eigenvalues and eigenvectors?

Eigenvalues and eigenvectors are important because they provide information about the behavior of a linear transformation. They can be used to understand how the transformation stretches, compresses, or rotates a vector in space. They also have applications in fields such as physics, engineering, and computer graphics.

3. How are eigenvalues and eigenvectors calculated?

Eigenvalues and eigenvectors can be calculated by solving the characteristic equation of a matrix, which is a polynomial equation that relates the eigenvalues to the entries of the matrix. The eigenvectors can then be found by solving a system of linear equations using the eigenvalues.

4. What is the relationship between eigenvalues and determinants?

The eigenvalues of a matrix are the roots of the characteristic equation, which is formed using the determinant of the matrix. This means that the determinant can be used to find the eigenvalues of a matrix. Additionally, the determinant of a matrix can be calculated using its eigenvalues.

5. How are eigenvalues and determinants used in real-world applications?

Eigenvalues and determinants have various applications in different fields. For example, in physics, they are used to describe the behavior of systems in quantum mechanics. In engineering, they are used in structural analysis and control systems. In finance, they are used in portfolio optimization and risk management. In computer science, they are used in image and signal processing.

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