Question about a Linear Algebra Proof?

In summary, the conversation discusses the relationship between eigenvalues and eigenvectors in a matrix. It is stated that if a matrix has n distinct eigenvalues, then it also has n distinct eigenvectors. The question is raised as to why, if the largest eigenvalue is 1 and the rest are less than or equal to 1, can we assume that any vector in the space can be written as a linear combination of the eigenvectors. It is clarified that this assumption is based on the eigenvectors forming a basis for the space. However, it is also noted that this does not necessarily mean that the coefficient c1 must be non-zero. The conversation concludes with a discussion on the meaning of "for any" in
  • #1
Punkyc7
420
0
If we have a matrix with n distinct eigenvalues I understand why we have n distinct eigenvectors.

My question is why is it if we have x[itex]_{1}[/itex]...x[itex]_{n}[/itex] eigenvectors of A with the largest eigenvalue equal to 1 and the rest of the eigenvalues are less than or equal to one, why can we assume that for any y[itex]_{0}[/itex] = c[itex]_{1}[/itex] x[itex]_{1}[/itex] + . . .+c[itex]_{n}[/itex] x[itex]_{n}[/itex]

for some constants c[itex]_{1}[/itex] ..c[itex]_{n}[/itex] with c[itex]_{1}[/itex] [itex]\neq[/itex]0.

Why does c[itex]_{1}[/itex] [itex]\neq[/itex]0.?

I am assuming that we can get any vector because the eigenvectors form a basis for the space we are in.
 
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  • #2
What are you actually trying to prove? Just because 1 is the largest eigenvalue there's no reason why the eigenvectors must be a basis. They might not be. And if you are given that x1..xn are a basis there is no reason why c1 must be nonzero. Suppose y0=x2?
 
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  • #3
I wasnt trying to prove anything I was trying to uderstand why the statement is written the way it is. The eigen vectors are linearly independent so since there is n of them doesn't that mean they form a basis?
 
  • #4
If an n by n matrix has n independent eigenvectors, then you can take the eigenvectors to be a basis (you can take any n independent vectors to be a basis for an n dimensional space).

But I don't understand what you mean by "for any y 0 = c 1 x 1 + . . .+c n x n".
What does the "for any" mean here? If you just mean that any vector in the space can be written as a linear combination of the basis vectors, of course that is true. But it is NOT true that c1 cannot be 0. For example, if y0 is any vector you can take y0= xn. In that case, cn= 1 while all other coefficients, including c1, are 0.
 

1. What is Linear Algebra?

Linear Algebra is a branch of mathematics that deals with linear equations, matrices, vectors, and vector spaces. It is used to solve problems in fields such as physics, engineering, economics, and computer science.

2. What is a proof in Linear Algebra?

A proof in Linear Algebra is a logical argument that shows a mathematical statement or theorem is true. It involves using established mathematical principles and rules to demonstrate the validity of a given statement.

3. How do you construct a proof in Linear Algebra?

To construct a proof in Linear Algebra, you first need to clearly state the theorem or statement that you are trying to prove. Then, you must justify each step in your argument using mathematical principles and rules. Finally, you must conclude with a statement that summarizes your proof and shows that the theorem is true.

4. Can you give an example of a proof in Linear Algebra?

One example of a proof in Linear Algebra is the proof of the invertible matrix theorem, which states that a square matrix is invertible if and only if its determinant is non-zero. The proof involves using the properties of determinants and matrix multiplication to show that if a matrix is invertible, its determinant must be non-zero, and vice versa.

5. Why are proofs important in Linear Algebra?

Proofs are important in Linear Algebra because they provide a rigorous and logical way to demonstrate the truth of mathematical statements. They also help to deepen our understanding of concepts and principles in Linear Algebra, and allow us to apply them in a variety of real-world problems and applications.

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