Is Every Scalar Multiple of an Eigenvector Also an Eigenvector?

[JayTheGent]

Hello PF, brand new member here.

A question about a proof:

If A*v=λ*v, then w = c*v is also an eigenvector of A.

This seems really simple to me, but perhaps I am doing it incorrectly:

A*c*v=λ*c*v, divide both sides by c and you are left with your original eigenvector of A. Am I missing something here?

 

[johnqwertyful]

Hello PF, brand new member here.

A question about a proof:

If A*v=λ*v, then w = c*v is also an eigenvector of A.

This seems really simple to me, but perhaps I am doing it incorrectly:

A*c*v=λ*c*v, divide both sides by c and you are left with your original eigenvector of A. Am I missing something here?

What you've shown is that if cv is an eigenvector, then v is an eigenvector. You need to show the other way. In addition, you can't always divide by c.

Also, this should be in homework help.

 

[WWGD]

notice that subspaces are closed under scaling. Or just factor out the c as the matrix cId.

 

[ehild]

Hello PF, brand new member here.

A question about a proof:

If A*v=λ*v, then w = c*v is also an eigenvector of A.

This seems really simple to me, but perhaps I am doing it incorrectly:

A*c*v=λ*c*v, divide both sides by c and you are left with your original eigenvector of A. Am I missing something here?

Welcome to PF!
Do not use * when applying an operator on a vector. The operator A assigns a vector u to vector v. Write u=A(v). In case v is an eigenvector of operator A, A(v)= λ*v. The right side is a product - a vector multiplied by a scalar, but the left-hand side is not.

You can use the property of linear operators that A(cv)=c A(v) (c is a scalar).

ehild

 

[HallsofIvy]

Hello PF, brand new member here.

A question about a proof:

If A*v=λ*v, then w = c*v is also an eigenvector of A.

This seems really simple to me, but perhaps I am doing it incorrectly:

A*c*v=λ*c*v, divide both sides by c and you are left with your original eigenvector of A. Am I missing something here?

As you have been told, what you shown is that "If cv is an eigenvector of A with eigenvalue λ then so is v". To prove the other way, reverse your proof: from Av= λv, multiply both sides by c.