# I Representation of a vector

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1. Aug 15, 2016

### 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?

2. Aug 15, 2016

### Staff: Mentor

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.

3. Aug 15, 2016

### Silviu

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?

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4. Aug 15, 2016

### Staff: Mentor

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.

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 $\mathfrak{su}(2)$.

Last edited: Aug 15, 2016
5. Aug 16, 2016

### Silviu

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).

6. Aug 16, 2016

### Staff: Mentor

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.

Last edited: Aug 16, 2016
7. Aug 18, 2016