Expressing a Matrix as a Linear Combination of Pauli Spin Matrices

[physics2000]

Homework Statement

Pauli Spin matrices (math methods in physics question)

Show that D can be expressed as:

$$D=d_1\sigma_1+d_2\sigma_2+d_3\sigma_3$$

and write the $$d_i$$ in terms of D's elements, let D also be Unitary

Homework Equations

- Any 2x2 complex matrix can be written as :

$$M=m_1\sigma_1+m_2\sigma_2+m_3\sigma_3+m_0I$$ where "I" is the identity matrix

- Pauli spin matrix properties

-require that D have 0 trace

The Attempt at a Solution

no idea where to begin honestly. please don't ding me ! This is the first day I've ever dealt with pauli spin matrices

 

[Dickfore]

If it is simply given that D is a unitary matrix, then this is not true. For example, take D to be a unit matrix (which is clearly unitary). Then, as you correctly point out, it is not traceless, and cannot be represented in that way.

What is D supposed to be?

If D is unitary, then it may be written in an exponential form:
$$D = \exp(i X)$$
What property does X have if D is unitary? What if D has a unit determinant?

 

[physics2000]

thanks for the reply,

thats exactly what I'm confused about...assume D is traceless and not unitary, does this make more sense to you? I get what you are saying, and I completely agree

I assume from the question that D is supposed to be like M in the relevant equations section, in which the $$\sigma_i$$ correspond to the pauli matrices 1,2 and 3

 

[George Jones]

If it is simply given that D is a unitary matrix, then this is not true. For example, take D to be a unit matrix (which is clearly unitary). Then, as you correctly point out, it is not traceless, and cannot be represented in that way.

Even though not every unitary 2x2 matrix is traceless, there are many unitary matrices that are tracekess, i.e., there are many unitary matrices that can be written in the form D of the original post. For example, each Pauli matrix is unitary and traceless. So is $i \left( \sigma_1 + \sigma_3) \right)/\sqrt{2}$. So is ...

 

[physics2000]

Thanks for the reply, I have no idea though how to answer the original question or start it. May I have some guidance :D?