Making a eigenvector a linear combination of other eigenvectors

[JordanGo]

Homework Statement

Write the eigenvector of $\sigma$x with +1 eigenvalue as a linear combination of the eigenvectors of M.

Homework Equations

$\sigma$x = (0,1),(1,0) (these are the columns)

The Attempt at a Solution

... Don't know what to do. Can someone show me how to do this using arbitrary eigenvectors, say (a,b) and (c,d)?

 

[clamtrox]

Can you find the eigenvector in question? Let's denote it by (v,w). Then you can write
(v,w) = C₁(a,b) + C₂(c,d) and solve the constant C's.

 

[JordanGo]

Ok, that makes sense, thanks a lot!

 

[HallsofIvy]

Homework Statement

Write the eigenvector of $\sigma$x with +1 eigenvalue as a linear combination of the eigenvectors of M.

Homework Equations

$\sigma$x = (0,1),(1,0) (these are the columns)

Okay, that's $\sigma$. What is M? we can't write something "as a linear combination of the eigenvectors of M without knowing what M is!

The Attempt at a Solution

... Don't know what to do. Can someone show me how to do this using arbitrary eigenvectors, say (a,b) and (c,d)?