Index notation of vector rotation

[shinobi20]

TL;DR

I'm a bit confused with the matrix notation and index notation of vector rotations. Specifically, how to understand the consistency of the index placement compared to the matrix notation.

Given a vector $\mathbf{r} = r^i e_i$ where $r^i$ are the components, $e_i$ are the basis vectors, and $i = 1, \ldots, n$. In matrix notation,

\begin{equation*}
\mathbf{r} = \begin{bmatrix} e_1 & e_2 & \ldots e_n \end{bmatrix} \begin{bmatrix} r^1 \\ r^2 \\ \vdots\\ r^n \end{bmatrix}
\end{equation*}

We can do an orthogonal transformation, e.g. rotation, on both the components and the basis vectors by inserting an identity matrix $I = R^T R$ in the middle and not change the vector. In index notation,

\begin{equation*}
\mathbf{r} = e_i (R^T)^i_{\; j} R^k_{\; i} r^i
\end{equation*}

Notice that the indices are placed in such a way that the summation with respect to the components and basis vectors makes sense. However, I'm a bit confused with the two notations,

\begin{align*}
& \mathbf{r} = e R^T R r \quad \text{matrix notation}\\
& \mathbf{r} = e_i (R^T)^i_{\; j} R^k_{\; i} r^i \quad \text{index notation}
\end{align*}

For the first equation, I believe $R^T R$ is a simple matrix multiplication so that without thinking of the current context it should follow the row-column rule for matrix multiplication $(R^T)^i_{\; j} R^j_{\; k}$. Notice that the column of $R^T$ labeled by $j$ is summed over the rows of $R$ also labeled by $j$. So, how do I reconcile this with the second equation?

I know that we could just switch the placement of $(R^T)^i_{\; j} R^k_{\; i}$ in the index notation since they are just numbers such that $(R^T)^i_{\; j} R^k_{\; i} =R^k_{\; i} (R^T)^i_{\; j}$; in this case the summed over indices is $i$ and it now follows the row-column summation having the same index. I think I have a partial understanding of the situation, but I think I need more clarification on this.

 

[PeroK]

TL;DR Summary: I'm a bit confused with the matrix notation and index notation of vector rotations. Specifically, how to understand the consistency of the index placement compared to the matrix notation.

Given a vector $\mathbf{r} = r^i e_i$ where $r^i$ are the components, $e_i$ are the basis vectors, and $i = 1, \ldots, n$. In matrix notation,

\begin{equation*}
\mathbf{r} = \begin{bmatrix} e_1 & e_2 & \ldots e_n \end{bmatrix} \begin{bmatrix} r^1 \\ r^2 \\ \vdots\\ r^n \end{bmatrix}
\end{equation*}

I don't think I've seen this idea before. I guess we are putting the basis vectors in a row matrix and calling that $e$, and putting the components of the vector $\mathbf r$ (in that basis) into a column matrix and calling that $r$. Then, the usual decomposition of the vector in that basis can be represented as a matrix multiplication: $er$.

We can do an orthogonal transformation, e.g. rotation, on both the components and the basis vectors by inserting an identity matrix $I = R^T R$ in the middle and not change the vector. In index notation,

\begin{equation*}
\mathbf{r} = e_i (R^T)^i_{\; j} R^k_{\; i} r^i
\end{equation*}

You can't have more that two indices the same in any summation. You need to use something else instead of one pair of the $i$'s.

Notice that the indices are placed in such a way that the summation with respect to the components and basis vectors makes sense. However, I'm a bit confused with the two notations,

\begin{align*}
& \mathbf{r} = e R^T R r \quad \text{matrix notation}\\
& \mathbf{r} = e_i (R^T)^i_{\; j} R^k_{\; i} r^i \quad \text{index notation}
\end{align*}

For the first equation, I believe $R^T R$ is a simple matrix multiplication so that without thinking of the current context it should follow the row-column rule for matrix multiplication $(R^T)^i_{\; j} R^j_{\; k}$. Notice that the column of $R^T$ labeled by $j$ is summed over the rows of $R$ also labeled by $j$. So, how do I reconcile this with the second equation?

I know that we could just switch the placement of $(R^T)^i_{\; j} R^k_{\; i}$ in the index notation since they are just numbers such that $(R^T)^i_{\; j} R^k_{\; i} =R^k_{\; i} (R^T)^i_{\; j}$; in this case the summed over indices is $i$ and it now follows the row-column summation having the same index. I think I have a partial understanding of the situation, but I think I need more clarification on this.

Putting in the extra matrices doesn't change the basic idea, as far as I understand it. Again, you can't have more than two indices the same.

 

[shinobi20]

I don't think I've seen this idea before. I guess we are putting the basis vectors in a row matrix and calling that $e$, and putting the components of the vector $\mathbf r$ (in that basis) into a column matrix and calling that $r$. Then, the usual decomposition of the vector in that basis can be represented as a matrix multiplication: $er$.

You can't have more that two indices the same in any summation. You need to use something else instead of one pair of the $i$'s.

Putting in the extra matrices doesn't change the basic idea, as far as I understand it. Again, you can't have more than two indices the same.

Oh! I see what's wrong. I should have written,

\begin{equation*}
\mathbf{r} = r'^i e'_i = e'_i r'^i = e_j (G^T)^j_{\; i} G^i_{\; k} r^k
\end{equation*}

Now, the matrix notation and the index notation match, is this correct?

Also, with regards to the idea that you have not seen, please take a peek at post.

 

[PeroK]

Oh! I see what's wrong. I should have written,

\begin{equation*}
\mathbf{r} = r'^i e'_i = e'_i r'^i = e_j (G^T)^j_{\; i} G^i_{\; k} r^k
\end{equation*}

Now, the matrix notation and the index notation match, is this correct?

Also, with regards to the idea that you have not seen, please take a peek at post.

It's the matrix notation that is new to me. I can see how it works, but it's stretching the concept of a matrix somewhat - your row matrix of basis vectors is not a row matrix of numbers, which is what you would normally expect. That said, I'm not sure what your question is.

 

[shinobi20]

It's the matrix notation that is new to me. I can see how it works, but it's stretching the concept of a matrix somewhat - your row matrix of basis vectors is not a row matrix of numbers, which is what you would normally expect. That said, I'm not sure what your question is.

I agree it is a valid concern, I think it is convenient to write it that way since we can think of a vector as the inner product of its components and basis vectors. Although, as to the precise mathematical justification, I'm not sure. One thing is that it is quite a common notation being used in the physics circle, I guess.

 

[PeroK]

I agree it is a valid concern, I think it is convenient to write it that way since we can think of a vector as the inner product of its components and basis vectors.

It's not really an inner product.

Although, as to the precise mathematical justification, I'm not sure. One thing is that it is quite a common notation being used in the physics circle, I guess.

Yes, I can see the temptation to write $[\mathbf e_1, \dots \mathbf e_n]$ and use that a "vector" of sorts. But, the physics texts I've seen have avoided this, in favour of using the Einstein summation notation.

 

[shinobi20]

It's not really an inner product.

Yes, I can see the temptation to write $[\mathbf e_1, \dots \mathbf e_n]$ and use that a "vector" of sorts. But, the physics texts I've seen have avoided this, in favour of using the Einstein summation notation.

I see. But talking about summation notation, is my rewrite in post #3 correct in terms of summation notation?

 

[PeroK]

I see. But talking about summation notation, is my rewrite in post #3 correct in terms of summation notation?

Okay, I think I see what you are trying to do. First, we can write down, with implied summation:
$$\mathbf r = r^i \mathbf e_i = r'^i \mathbf e'_i$$That's just a vector expressed in two different bases.

Now, the primed and unprimed bases must be related. I.e. each basis vector itself can be expressed in the other basis. In general, we would have something like:
$$\mathbf e_i = T_i^j \mathbf e'_j$$Where, at this stage, the $T_i^j$ are just a set of numbers. You can then put these numbers into a matrix. We also have another set of numbers $S_i^j$, where:
$$\mathbf e'_i = S_i^j \mathbf e_j$$Note that, in general:
$$\mathbf e_i = T_i^jS_j^k \mathbf e_k$$And we can see that the matrices $T$ and $S$ are inverses of each other.

What you are doing is using these transformation matrices informaly to write this vector equation in an informal matrix format:
$$e = Te'$$, where $e$ and $e'$ are row and column "matrices" with the bases vectors as entries. That's purely notational and I'm not sure what the problem is.

What's important is that we often want to know how the components of a vector transform under a change of basis. I.e. we want to relate the components $r^i$ to the components $r'^i$. We can see that:
$$\mathbf r = r^i \mathbf e_i = r'^j \mathbf e'_j = r'^j S_j^i \mathbf e_i$$And, by uniqueness of the expansion in a given basis, we can see that:
$$r^i = r'^j S_j^i = (S^T)^i_jr'^j$$In other words, if we want to use this in matrix form, we have to use the transpose of the matrix $S$.

This is what's tricky. We are using the "other" matrix, transposed. In particular, the components of a vector transform differently from the basis vectors themselves.

That's the important thing. The rest is notational.