Understanding Eigenvalues and Determinants with Repeated Multiplicities

[Jennifer1990]

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 ~~

 

[Dick]

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).

 

[Jennifer1990]

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

 

[Dick]

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)?

 

[Jennifer1990]

How can i write our the characteristic polynomial if we're dealing with a general nxn matrix?

 

[Dick]

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.

 

[Jennifer1990]

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

 

[Dick]

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?

 

[Jennifer1990]

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?

 

[Dick]

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.