Choosing coefficients for eigenvectors.

[Lavabug]

Homework Statement

For the following operator represented in the orthonormal basis {|1>, |2>}

$$\hat{M} = \begin{pmatrix} 2 & i\sqrt{2} \\ -i\sqrt{2 & 2} \end{pmatrix}$$

find the eigenvalues and eigenvectors and express them as a function of |1> and |2> normalized.

The Attempt at a Solution

I'm trying to apply what I learned way back in linear algebra. When I solve for the eigenvector coefficients, I end up with a system of 2 variables and 1 equation (alternatively 2 variables and a rank 1 matrix), which gives me (2 - 1) free parameters to set arbitrarily. However back in algebra I used to just pick a variable "gamma" or whatever and leave my eigenvector in terms of it. Now I'm supposed to fix an actual number to it? For what reason and how should I pick my free parameters?

 

[vela]

Homework Statement

For the following operator represented in the orthonormal basis {|1>, |2>}

$$\hat{M} = \begin{pmatrix} 2 & i\sqrt{2} \\ -i\sqrt{2} & 2 \end{pmatrix}$$

find the eigenvalues and eigenvectors and express them as a function of |1> and |2> normalized.

The Attempt at a Solution

I'm trying to apply what I learned way back in linear algebra. When I solve for the eigenvector coefficients, I end up with a system of 2 variables and 1 equation (alternatively 2 variables and a rank 1 matrix), which gives me (2 - 1) free parameters to set arbitrarily. However back in algebra I used to just pick a variable "gamma" or whatever and leave my eigenvector in terms of it. Now I'm supposed to fix an actual number to it? For what reason and how should I pick my free parameters?

Why? Because the problem is asking you to.

Just arbitrarily set the free parameter to, say, 1, and then normalize the resulting vector.

 

[Lavabug]

Why? Because the problem is asking you to.

Just arbitrarily set the free parameter to, say, 1, and then normalize the resulting vector.

Thanks. Today we went over an identical problem in class, what we did was basically write down the eigenvector as a function of only 1 coefficient, then impose the orthonormalization condition to solve for it to get the normalized eigenvector.

I forgot to mention the other part of the problem:

Write down in matrix form the projectors over the 2 normalized eigenvectors. Verify that they satisfy the closure and orthogonality relations.
Last one seems straightforward, just check that my 2 eigenvectors satisfy $\langle\Psi_{1N}|\Psi_{2N}\rangle = 0$.

But I have no idea how the first thing is done. From my notes it looks like we wrote down the unit matrix in the form of dot products between the unit vectors, and stuck the projection operator in between the dot products for each.

Checking for closure I understand is just adding up the projection operators to see if they give the identity matrix, but I need to understand the previous part of the question.