Eigen value 0 prove

  • #1
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prove that a matrix cannot be invertible if it has eigen value of 0.

not invertible mean that the determinant equals 0.

x(x-2)=0
??
 

Answers and Replies

  • #2
33,712
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prove that a matrix cannot be invertible if it has eigen value of 0.

not invertible mean that the determinant equals 0.

x(x-2)=0
??
Where does x(x -2) come in?

You have an n x n matrix A that has 0 as an eigenvalue. How do you find the eigenvalues of a matrix? How are eigenvalues defined?
 
  • #3
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x(x -2) its just an example for a polynomial which is not inertible
i thought of proving like this

that if an operator T is not invertible then det T=0 then dim ker differs 0
(we have a row of zeros) so i have eigen value 0

correct
??
 
Last edited:
  • #4
1,395
0
Where does x(x -2) come in?

You have an n x n matrix A that has 0 as an eigenvalue. How do you find the eigenvalues of a matrix? How are eigenvalues defined?
"If an equation containing a variable parameter possesses nontrivial solutions only for certain special values of the parameter, these solutions are called eigenfunctions and the special values are called eigenvalues."

how to use it
??
 
  • #5
Office_Shredder
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Not the kind of eigenvalue you want (well, sort of is, but let's not complicate things). An eigenvalue [tex] \lambda [/tex] of a matrix A is a number such that there exists v in the vector space, v non-zero, with

[tex]Av = \lambda v[/tex]
 
  • #6
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so if i have eigen value zero
Av=0

how does it show that its not invertible
i know that now v plays the role of kernel A
but i dont know what is the link between the kernel of an operator and its
ability to be invertible
??
 
  • #7
Office_Shredder
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A matrix is invertible if it has... an inverse. To have an inverse, it must be 1-1. Think about what having a non-trivial kernel means with regards to this
 
  • #8
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1-1 means that for each x i have a y value and vise versa
i really dont know what is the affect of kernel
i know how to find it in a matrix
i know
dim Im + dim Ker =V

??
 
  • #9
statdad
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Remember that in order for a vector [tex] v [/tex] to be an eigenvector, it must be true that [tex] v \ne 0[/tex]. If that is the case, yet [tex] A v = 0 [tex], what can you conclude?
 
  • #10
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zero vector is the product of multiplication of the operarator by the kernel vectors

i dont know
??
 
  • #11
statdad
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Well, remember that if the matrix [tex] A [/tex] is invertible, the solution to

[tex]
Ax = b
[/tex]

is always given by

[tex]
x = A^{-1}b
[/tex]

Now, suppose that [tex] v \ne 0 [/tex] is an eigenvector that corresponds to the eigenvalue [tex] \lambda = 0 [/tex]

Now assume that [tex] A [/tex] is invertible. What does the equation

[tex]
Av = \lamba \, v = 0
[/tex]

tell you about [tex] v [/tex]? What does this contradict, and what assumption led to that contradiction?
 
  • #12
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573
Can you use the fact that the inverse is the matrix of cofactors divided by the determinant, and that division would be undefined if you had a zero eigenvalue because the determinant is the product of the eigenvalues?
 
  • #13
655
3
Consider
Ax = 0
A0 = 0

What is A-1(0)?
 
  • #14
1,395
0
A matrix is invertible if it has... an inverse. To have an inverse, it must be 1-1. Think about what having a non-trivial kernel means with regards to this
what is the answer
??
 
  • #15
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why if dim ker differs 0
then its not invertible
??
 
  • #16
33,712
5,412
Let's get organized. You're trying to prove this statement, which is a paraphrase of your first post in this thread:

If matrix A has 0 as an eigenvalue, A does not have an inverse.​

If A has 0 as an eigenvalue, Ax = 0x = 0, for some nonzero vector x (the eigenvector associated with eigenvalue 0).

So Ax = 0, where x is not the zero vector.
From this equation, is it possible to say either that A definitely has an inverse or that A definitely doesn't have an inverse?
 
  • #17
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i can say that definitely we have a kernel vector in A
but i dont know how it affects the invertiblity of A
??
 
  • #18
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There is always at least one vector in ker(A), regardless of whether A has an inverse.
 
  • #20
HallsofIvy
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This has already been said, but I'll try again. If Ax= 0 and A has an inverse, what do you get if you multiply both sides of that equation by A-1? What does that tell you about the kernel of A? How is that a contradiction?
 
  • #21
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i get x=0
what does it mean?
 
  • #22
HallsofIvy
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That means that "x= 0" is the only solution to Ax= 0. What does that tell you about the kernel of A?
 

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