No problem, it's always good to have multiple sources!

[Silviu]

Hello. If I represent a vector space using matrices, for example if a 3x1 vector, V, is represented by 3x3 matrix, A, and if this vector was the eigenvector of another matrix, M, with eigenvalue v, if I apply M to the matrix representation of this vector, does this holds: MA=vA? Also, if I represent the vector by a n x n matrix, how do I transform the matrix M, such that the equation of eigenvector still holds?

 

[fresh_42]

Hello. If I represent a vector space using matrices...

What do you mean by this? Do you mean writing a vector $v=\sum_{\iota \in I}v_\iota b_ \iota$ as $v=(..., v_\iota , ...)$ in coordinate form according to a basis $\{b_\iota\}_{\iota \in I}$?

..., for example if a 3x1 vector, V, is represented by 3x3 matrix, A, ...

A vector is something which a matrix applies to: $v \mapsto Av$. Of course you may take $v$ as a row or column vector of $A$ and fill up the rest with zeroes, but why? $v$ and $A$ are different objects, one is something that make up vector spaces and the other one is a mapping between vector spaces.

...and if this vector was the eigenvector of another matrix, M, with eigenvalue c [changed by me], ...

which means $Mv = c\, v$.

... if I apply M to the matrix representation of this vector, ...

So your vector $v$ (with the dimensions above) is a $(3\times 1)-$matrix and $M$ a $(3\times 3)-$matrix.

... does this holds: MA=cA? [changed by me]

This depends on how you "represent" $v$ by $A$. If $A = (v,0,0)$ then of course. But why should you do this?

Also, if I represent the vector by a n x n matrix, how do I transform the matrix M, such that the equation of eigenvector still holds?

Again. There is no meaning in "representing" a vector as a square matrix, unless in very special cases (which I can't imagine). The only natural way is to see a vector as a $(n \times 1)-$matrix.
My personal opinion is, that you should forget about it and recapture what vectors and linear mappings are. They are not supposed to be messed up.

 

[Silviu]

What do you mean by this? Do you mean writing a vector $v=\sum_{\iota \in I}v_\iota b_ \iota$ as $v=(..., v_\iota , ...)$ in coordinate form according to a basis $\{b_\iota\}_{\iota \in I}$?A vector is something which a matrix applies to: $v \mapsto Av$. Of course you may take $v$ as a row or column vector of $A$ and fill up the rest with zeroes, but why? $v$ and $A$ are different objects, one is something that make up vector spaces and the other one is a mapping between vector spaces.which means $Mv = c\, v$.So your vector $v$ (with the dimensions above) is a $(3\times 1)-$matrix and $M$ a $(3\times 3)-$matrix.This depends on how you "represent" $v$ by $A$. If $A = (v,0,0)$ then of course. But why should you do this?Again. There is no meaning in "representing" a vector as a square matrix, unless in very special cases (which I can't imagine). The only natural way is to see a vector as a $(n \times 1)-$matrix.
My personal opinion is, that you should forget about it and recapture what vectors and linear mappings are. They are not supposed to be messed up.

Hello! Thank you for your answer! The reason why I am confused is what I read in a book about SU(2) representation. I attached a part of what I read. They aim to "reduce the Hilbert space of the world to block diagonal form". I am confused as the Hilbert space contains vectors, but block diagonal is related to matrices. This is why I assumed they try to represent vectors by square matrices... Do they mean something else there?

 

[fresh_42]

Well, this is a very general and introductory text. I've answered a similar question about $SU(3)$ yesterday. You may read it and see where confusion may come from. It also might help to align what's written in your quote.

https://www.physicsforums.com/threads/bases-for-su-3-adjoint-representation.881928/#post-5543245

If you substitute $SU(3)$ by $SU(2)$, $\mathbb{C}^3$ by $\mathbb{C}^2$ and Gell-Mann matrices by Pauli-matrices plus of course another dimension ($3$ instead of $8$), it applies to your situation likewise.

Edit: ... and of course $\mathfrak{su}(3)$ by $\mathfrak{su}(2)$.

 

[Silviu]

Well, this is a very general and introductory text. I've answered a similar question about $SU(3)$ yesterday. You may read it and see where confusion may come from. It also might help to align what's written in your quote.

https://www.physicsforums.com/threads/bases-for-su-3-adjoint-representation.881928/#post-5543245

If you substitute $SU(3)$ by $SU(2)$, $\mathbb{C}^3$ by $\mathbb{C}^2$ and Gell-Mann matrices by Pauli-matrices plus of course another dimension ($3$ instead of $8$), it applies to your situation likewise.

Edit: ... and of course $\mathfrak{su}(3)$ by $\mathfrak{su}(2)$.

Hello! Thank you for this, but I am still a bit confused. They say "the states of the representation can be written as (j, m)". What do they mean by the state of a representation? I thought that the state (j, m) is an element of the vector space, while the representation is a matrix representation of the group of operations acting on this vector space. But, by what they are saying, I understand that they represent the vector space itself, and I am a bit confused why would you do that in general? (I understood the reason for representations of the group of operations, but I can't understand yet the reason of representing the vector space where you apply the operations).

 

[fresh_42]

Hello! Thank you for this, but I am still a bit confused. They say "the states of the representation can be written as (j, m)". What do they mean by the state of a representation? I thought that the state (j, m) is an element of the vector space, while the representation is a matrix representation of the group of operations acting on this vector space. But, by what they are saying, I understand that they represent the vector space itself, and I am a bit confused why would you do that in general? (I understood the reason for representations of the group of operations, but I can't understand yet the reason of representing the vector space where you apply the operations).

I have found a description of this connection (between math and physics language) on the example of Pauli matrices here. It's in the wrong language, but you could either ignore the texts and only regard the formulas or take it as bad written English since the important words are almost the same. (I've linked to the relevant section, so it's not necessary to consider the entire page. It's been easier than to retype all formulas.) I haven't looked into deep but I hope it can help you. Also the English version of this page (which isn't simply a translation) could be interesting to read.

 

[jim mcnamara]

Does the English version help?

https://en.wikipedia.org/wiki/Pauli_matrices

...oops ignore my post fresh_42 has it already posted.