Understanding Choi's Theorem and Notation in Matrix Theory

[naima]

I am reading the proof of the Choi's theorem in his own paper.
he first introduces $E_{ij}$ as the nn null matrix but with a 1 at i,j.
Then he uses $(E_{ij})_{1<=i,j <=n}$
he says that thi is a positive matrix. What is talking about?
Is it a matrix of matrices?

 

[jedishrfu]

I think the notation means that 1<=i<=n and 1<=j<=n ie for an n x n matrix the i and j indices can have integer values between 1 and n.

 

[fresh_42]

I am reading the proof of the Choi's theorem in his own paper.
he first introduces $E_{ij}$ as the nn null matrix but with a 1 at i,j.
Then he uses $(E_{ij})_{1<=i,j <=n}$
he says that thi is a positive matrix. What is talking about?
Is it a matrix of matrices?

I read this as $E_{kl} = (\delta_{ki}\delta_{jl})_{i,j}$ and $(E_{ij})_{1<=i,j <=n}$ as either an all one matrix or more likely the same as $E_{ij}$ with only the ranges of $i$ and $j$ added, as he considers non square matrices as well. A standard basis vector if you like.

 

[naima]

There is no problem with the first definition. it is a n*n matrix with one "1".
Do you agree that the second is a n^2 * n^2 with n^2 "1".in it?

 

[fresh_42]

Do you agree that the second is a n^2 * n^2 with n^2 "1".in it?

This would really surprise me. I think it is more like an ill-fated version of $A=(a_{ij})_{1≤i≤n,1≤j≤n}$. However, I wouldn't bet on it.
What could be a reason to arrange the $E_{ij}$ like this, as a matrix of matrices?

 

[naima]

I do not feel comfortable with the proof of the Choi's theorem.
But read the top of page 2. we have $M_n(M_m)$ which is a tensor product.
Choi says that this space contains
"n X n block matrices with m x m matrices as entries"

 

[fresh_42]

I do not feel comfortable with the proof of the Choi's theorem.
But read the top of page 2. we have $M_n(M_m)$ which is a tensor product.
Choi says that this space contains
"n X n block matrices with m x m matrices as entries"

Yes, tensors can viewed this way:
$c \otimes d = cd$ with scalars $c,d$ is a scalar.
$c \otimes v$ with $c$ a scalar and $v$ a vector is a vector.
$v \otimes w$ with $v,w$ vectors is a matrix (of rank 1).
$v \otimes A$ with a vector $v$ and a matrix $A$ is a stack of weighted copies of $A$.
$A \otimes B$ with matrices $A,B$ are a four-dimensional array of coordinates.
... and so on ...
The rest of tensor spaces are linear combinations of those.

It may be right that $(E_{i,j})_{i,j}$ is a tensor product or otherwise arranged array of matrices. I don't want to rule it out. I simply haven't seen a construction like this noted in coordinates. It would be a linear function of linear functions, and all in coordinates.

 

[naima]

In this http://isites.harvard.edu/fs/docs/icb.topic1533461.files/Chois%20Theorem-Bill.pdf

 

[fresh_42]

In this http://isites.harvard.edu/fs/docs/icb.topic1533461.files/Chois%20Theorem-Bill.pdf

I don't know whether it is always like that. Rui Li's notations are new to me. E.g. I see $A\otimes B$ as a four-dimensional array, but this can't be put on paper. So he writes $A \otimes B = (A_{ij}B)_{ij}$. This makes certainly sense though.
But I haven't heard of a generalized or partial trace as in $A.3$. However, I'm not a physicist and a little bit allergic to coordinates, because they often hide the principle behind. And furthermore: it isn't important whether it is always the case or not, because Rui Li defines his notations and is consistent.