Is Lambda an Eigenvalue of A in the Cayley-Hamilton Theorem Proof?

[vabamyyr]

i met a proof to cayley hamilton theorem and have some questions.

It uses that lambda*I - A is invertible. But lambda is surely an eigenvalue of A and 1/(lamda*I - A) is not legit so how is it legal to use that.

 

[matt grime]

Writing 1/(lambda*I-A) is also not allowed.

Why is lambda an eigenvalue? Who says so? It is just a greek letter, probably representing some scalar. As it is unles you post all of the proof who can possibly say whether it is correct or not.

 

[vabamyyr]

http://www.math.chalmers.se/~wennberg/Undervisning/ODE/linalg.pdf

 

[vabamyyr]

Also I have some questions on these topics

 

[Muzza]

The first sentence of the proof specifically states that "if lambda is not an eigenvalue of A"...

 

[0rthodontist]

I don't know about cayley-hamilton but I do know that lambda is an eigenvalue of A iff lambda * I - A is NOT invertible.

 

[matt grime]

Hmm? What do you mean by that (in regards to this post)?

 

[0rthodontist]

Ah, I misinterpreted his post. At first reading I thought he was claiming that lambda * I - A is invertible meant that lambda was an eigenvalue of A. Now I see that he was claiming lambda was an eigenvalue of A separately from that statement.