What Defines Mainvectors in the Context of Eigenvalue Multiplicities?

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In summary: The equation you are solving is the generalized eigenvalue equation. So by solving that equation you can find a basis of eigenvectors.
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yeus
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When there is a difference between algebraic and geometric multiplicity of eigenvalues mainvectors are used to handle that difference. Mainvectors are defined as the solution v of the equation: (A-lambda*E)^k*v=0 where k is the multiplicity of the eigenvalue lambda. Now my question is: Why are you using the k-th power of the definition of an eigenvector to search for a mainvector? how do you get from the fact that there is a difference between alge./geom. mult. to that equation? thanks for answers

(I apologize for not knowing that many mathematical expressions in english. but I hope you guys understand my problem)
 
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so sad... no one wants to answer my question ;) please guys
 
  • #3
Perhaps I can exlpain why there have been no answers:

google for "mainvector" and see how many hits you get, or at least how many are to do with mathematics.

One confusing point is that you say this is to do with the diffence between algebraic and geometric mutliplicites but then state k is "the multiplicity". Well, whcih is it? Algebraic or geometric?

Why not run through it with the example of the matrix

11
01

so that the algebraic multiplicity of the eigenvalue 1 is 2 but its geometric multiplicty is 1.

Should k be one or two?

I think you may need the word principal instead of main, and that what you're getting at is the difference between an eigenspace and a generalized eigenspace.

If A is a liner map and t an eigen value, then an eigenvector is a nonzero vector such that (A-t)v=0. Sometimes we can find a basis of eigenvectrs, but usually not. The next best thing we can do is, instead of diagonalizing the matrix, put it into Jordan Form, say by choosing a break down of the vector space into *generalized eigenspaces*, that is a set of vectors such that (A-t)^k vanishes on it for some k. We can then write V is a direct sum of subspaces where A acts as


t10000...
0t1000...
00t100...
000t10...
.
.
.
 
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  • #4
matt grime said:
The next best thing we can do is, instead of diagonalizing the matrix, put it into Jordan Form, say by choosing a break down of the vector space into *generalized eigenspaces*, that is a set of vectors such that (A-t)^k vanishes on it for some k. We can then write V is a direct sum of subspaces where A acts as


t10000...
0t1000...
00t100...
000t10...
.
.
.

well.. first of all thanks for you answer... it helped me a bit, but the thing in the qotation marks, -thats exactly the part that i don't understand- why can you find a basis of eigenvecotrs in the generalized eigenspace? I may have a bug in my brain regarding to this forgive me ;)
 

Related to What Defines Mainvectors in the Context of Eigenvalue Multiplicities?

What is the definition of mainvectors?

Mainvectors are a set of vectors that form the basis for a vector space. They are linearly independent and can be used to represent any vector within the vector space through a unique combination of scalar multiples.

Why are mainvectors important?

Mainvectors are important because they allow us to understand the structure and properties of a vector space. They also provide a way to simplify vector calculations and make them more efficient.

How do you find the mainvectors of a vector space?

The mainvectors of a vector space can be found by using the Gram-Schmidt process. This involves taking a set of linearly independent vectors and transforming them into orthogonal vectors, which can then be used as the mainvectors.

Can a vector space have more than one set of mainvectors?

Yes, a vector space can have multiple sets of mainvectors. However, each set must have the same number of vectors and they must all be linearly independent. This means that the sets will span the same vector space, but they may represent it in different ways.

What is the difference between mainvectors and basis vectors?

Mainvectors and basis vectors are often used interchangeably, but there is a slight difference. Mainvectors are a specific set of vectors that form the basis for a vector space, while basis vectors can refer to any set of linearly independent vectors that span a vector space.

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