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

In summary, the conversation discusses the use of lambda as an eigenvalue in the proof of the Cayley-Hamilton theorem. The speaker has questions about the legitimacy of using 1/(lambda*I-A) and the validity of lambda as an eigenvalue. They also refer to a document for further understanding of the topic.
  • #1
vabamyyr
66
0
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.
 
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  • #2
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.
 
  • #3
http://www.math.chalmers.se/~wennberg/Undervisning/ODE/linalg.pdf
 
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  • #4
Also I have some questions on these topics
 

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  • #5
The first sentence of the proof specifically states that "if lambda is not an eigenvalue of A"...
 
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  • #6
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.
 
  • #7
Hmm? What do you mean by that (in regards to this post)?
 
  • #8
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.
 

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

Question 1: What is the Cayley-Hamilton Theorem?

The Cayley-Hamilton Theorem states that every square matrix satisfies its own characteristic equation. This means that if you substitute the matrix itself into its own characteristic equation, the result will always be equal to the zero matrix.

Question 2: Who were Cayley and Hamilton?

Arthur Cayley and William Hamilton were two mathematicians who independently discovered and proved the Cayley-Hamilton Theorem in the mid-19th century.

Question 3: How is the Cayley-Hamilton Theorem used in linear algebra?

The Cayley-Hamilton Theorem is a fundamental result in linear algebra and is used in various applications, including solving systems of linear equations, finding eigenvalues and eigenvectors, and diagonalizing matrices.

Question 4: Can the Cayley-Hamilton Theorem be extended to non-square matrices?

No, the Cayley-Hamilton Theorem only applies to square matrices. However, there are similar theorems that apply to non-square matrices, such as the Jordan Canonical Form Theorem.

Question 5: What are some real-world applications of the Cayley-Hamilton Theorem?

The Cayley-Hamilton Theorem has applications in many fields, including physics, engineering, and computer science. It is used in control theory, signal processing, and image processing, among others.

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