[Linear Algebra] Conjugate Transpose of a Matrix and vectors in ℂ

[Ismail Siddiqui]

Homework Statement

Let A be an n x n matrix, and let v, w ∈ ℂⁿ.

Prove that Av ⋅ w = v ⋅ A^†w

Homework Equations

† = conjugate transpose
⋅ = dot product
* = conjugate
T = transpose

(AB)^-1 = B^-1A^-1
(AB)^-1 = B^TA^T
(AB)^* = A^*B^*
A^† = (A^T)^*
Definitions of Unitary and Hermitian Matrices
Complex Mod
Vector Inner Product Space rules/axioms
Cancellation Theorem

The Attempt at a Solution

(Av) ⋅ w = v ⋅ (A^†w)
(Av) ⋅ w = ((A^†w) ⋅ v)^*
(Av) ⋅ w = ((A^†w)^*) ⋅v^*
(Av) ⋅ w = ((A^†)^*w^*) ⋅v^*
(Av) ⋅ w = (A^Tw^*) ⋅v^*

and that's as far as I get :)

 

[micromass]

How is $\mathbf{v}\cdot \mathbf{w}$ defined? Can you write this as a matrix multiplication?

 

[andrewkirk]

Write the LHS out as a double sum of matrix and vector components and rearrange until it is equal to the RHS.

There may be a quicker way but we can't really suggest anything without knowing what theorems you are allowed to use.

 

[Ismail Siddiqui]

@micromass The LHS and RHS of the equations are defined by the vector dot product, I've made some changes in the original post in an attempt to clarify that.

@andrewkirk I've added more to the relevant equations sections to address what you said. There are probably a few more theorems but this is what I was able to pull off the top of my head.

Thank you!

 

[micromass]

What is the definition of the vector dot product?

 

[andrewkirk]

@andrewkirk I've added more to the relevant equations sections to address what you said. There are probably a few more theorems but this is what I was able to pull off the top of my head.

Those won't enable you to get a solution because none of them address the critical issue of what happens when you move an operator from the first part of an inner product to the second.
Write the equation out in component form. It's only a few lines to prove it that way, and no external theorems need to be used.

 

[Ismail Siddiqui]

What is the definition of the vector dot product?

v ⋅ w = ∑ v_iw_i where v_i and w_i are the ith entries of the vectors v and w and 1 ≤ i ≤ n where n is the last index.

 

[micromass]

v ⋅ w = ∑ v_iw_i where v_i and w_i are the ith entries of the vectors v and w and 1 ≤ i ≤ n where n is the last index.

That is not the definition of the dot product since you work in $\mathbb{C}^n$.

If you do find the correct definition, are you able to write it as a matrix multiplication?

 

[Ismail Siddiqui]

Those won't enable you to get a solution because none of them address the critical issue of what happens when you move an operator from the first part of an inner product to the second.
Write the equation out in component form. It's only a few lines to prove it that way, and no external theorems need to be used.

I'll try it that way and see where I get, thanks.

 

[andrewkirk]

I'll try it that way and see where I get, thanks.

Make sure you use the correct definition of the inner product in a complex vector space, as per @micromass's post above. Otherwise it won't work out.

 

[Ismail Siddiqui]

That is not the definition of the dot product since you work in $\mathbb{C}^n$.

If you do find the correct definition, are you able to write it as a matrix multiplication?

oh boy, your absolutely right. I completely forgot to add on the complex conjugate. It should be ∑v_i* w_i.

 

[Ismail Siddiqui]

@andrewkirk @micromass I wrote it out in component form and re-arranged it from there. Worked out perfectly.Thanks again!