# Tsampirlis Chapter 1 Inner Product

1. Feb 16, 2016

### Waxterzz

Hi all,

The basis vectors are defined as 1x3 matrices, how can the result be a 3x3 matrix?

How can the result of a dot product be a 3x3 matrix, I'm stumbled, how can I evaluate this?

Thanks.

2. Feb 16, 2016

### Samy_A

You have 9 dot products.
You have a term $g_{\mu\nu}$ for all the possible combinations of $\mu, \ \nu$. That's 3*3.

The 3*3 matrix is :
$\begin{pmatrix} e_1\cdot e_1 & e_1\cdot e_2 & e_1\cdot e_3\\ e_2\cdot e_1 & e_2\cdot e_2 & e_2\cdot e_3\\ e_3\cdot e_1 & e_3\cdot e_2 & e_3\cdot e_3 \end{pmatrix}$

3. Feb 16, 2016

### Waxterzz

Apparently I have never heard of a matrix of an inner product.

4. Feb 16, 2016

### Waxterzz

But in the book they define lower indices as the number of colums, so I thought it were two 1x3 matrices?

5. Feb 16, 2016

### Samy_A

An inner product associates a pair of vectors with a scalar (and has a number of properties that I won't write down here).
So with a basis of 3 vectors, you can associate 3*3 inner products, and that gives you the matrix I posted above.
I don't know about the notation in your book, but in the image you posted it is clearly stated that $e_1, e_2, e_3$ form a basis, thus each of them is a vector. Their respective inner products is a perfectly well defined scalar.

I only watch the beginning of the video, but yes, the matrix representation of a inner product she is computing is the same concept as the one in your book. She uses a somewhat different notation, though.

6. Feb 16, 2016

### Waxterzz

So suddenly the lower indices represent columns in the case of μ and rows in the case for ν? And it's just plain matrix multiplication.
First thing Samparlis did was define this in the book:

7. Feb 16, 2016

### Waxterzz

So when I saw eu and ev I thought two 1x3 matrices, since see notation above, which is not right, right?

8. Feb 16, 2016

### Waxterzz

So my error is, I don't understand the notations? Lower indices are not always columns?

If he defines the upper indices as rows and columns lower indices why isn't it

guv = eu . ev

instead of writing everything lower index

9. Feb 16, 2016

### Samy_A

Forget the notation for a moment. (I'm looking at the book right now, trying to understand it.)
More important: do you understand how the combination of a basis (3 vectors) and an inner product leads to the definition of the matrix representation of that inner product?

10. Feb 16, 2016

### Waxterzz

Apparently not, but in the video I posted it is defined as 2x2 matrices and I don't have an example for a 3x3 matrix.

I have notion of Linear Algebra, never encountered matrix representation of an inner product.

11. Feb 16, 2016

### Waxterzz

12. Feb 16, 2016

### Samy_A

$e_1, e_2, e_3$ are (basis) vectors. They are not 1*3 matrices. The 1*3 matrix is $(e_1, e_2, e_3)$, the matrix consisting of the three vectors taken together.
There is nothing very difficult with this notion.
If you get it in two dimensions, it is conceptually the same in 3 or 4 or 10 dimensions.

He defines $g_{\mu\nu}$ as the inner product of the vectors $e_\mu$ and $e_\nu$.

13. Feb 16, 2016

### Waxterzz

So what he meant is

e1
e2 times e1 e2 e3
e3

and ordinary matrix multiplication?

14. Feb 16, 2016

### Samy_A

You can represent it this way. Just keep in mind that $e_1,e_2,e_3$ are themselves vectors, not scalars. And that the "product" of two vectors is the inner product.

15. Feb 19, 2016

### Waxterzz

To come back on my question.

Part of the confusion arose, because I've forgotton or looked over the meaning of the ≡ character. They were referring to the elements of the matrix. Stupid of me, but it's clear now.