Tsampirlis Chapter 1 Inner Product

In summary: He defines ##g_{\mu\nu}## as the inner product of the vectors ##e_\mu## and...He defines ##g_{\mu\nu}## as the inner product of the vectors ##e_\mu## and ##e_u##.The notation ##g_{\mu\nu}## is just a shorthand way of writing the three equations above.
  • #1
Waxterzz
82
0
Hi all,

tyfAGOW.jpg


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?

A inner product returns a scalar, and now it returns a 3x3 matrix, please help.

Thanks.
 
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  • #2
Waxterzz said:
Hi all,

tyfAGOW.jpg


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?

A inner product returns a scalar, and now it returns a 3x3 matrix, please help.

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



Should I follow this?
 
  • #4
Samy_A said:
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}##
But in the book they define lower indices as the number of colums, so I thought it were two 1x3 matrices?
 
  • #5
Waxterzz said:
But in the book they define lower indices as the number of colums, so I thought it were two 1x3 matrices?
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
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:

nIX9L4j.jpg
 
  • #7
Waxterzz said:
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:

nIX9L4j.jpg

So when I saw eu and ev I thought two 1x3 matrices, since see notation above, which is not right, right?
 
  • #8
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 itguv = eu . ev

instead of writing everything lower index
 
  • #9
Waxterzz said:
So my error is, I don't understand the notations? Lower indices are not always columns?
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
Samy_A said:
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?
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.
 
  • #12
Waxterzz said:
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:

nIX9L4j.jpg
##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.
Waxterzz said:
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.
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.

Waxterzz said:
If he defines the upper indices as rows and columns lower indices why isn't itguv = eu . ev

instead of writing everything lower index
He defines ##g_{\mu\nu}## as the inner product of the vectors ##e_\mu## and ##e_\nu##.
 
  • #13
So what he meant is

e1
e2 times e1 e2 e3
e3

and ordinary matrix multiplication?
 
  • #14
Waxterzz said:
So what he meant is

e1
e2 times e1 e2 e3
e3

and ordinary matrix multiplication?
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
Samy_A said:
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.
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.
 

What is an inner product?

An inner product is a mathematical operation that takes two vectors and produces a scalar value. It is often denoted as ⟨u, v⟩ and is used to measure the angle between two vectors, the length of a vector, and the projection of one vector onto another.

How is the inner product calculated?

The inner product is calculated by taking the dot product of the two vectors. This means multiplying the corresponding components of the vectors and adding them together. For example, if u = [1, 2, 3] and v = [4, 5, 6], then the inner product would be ⟨u, v⟩ = 1 * 4 + 2 * 5 + 3 * 6 = 32.

What are some properties of the inner product?

Some properties of the inner product include linearity, symmetry, and positive definiteness. Linearity means that the inner product of two vectors multiplied by a scalar is equal to the same scalar multiplied by the inner product of the two vectors. Symmetry means that the order of the vectors does not matter, and positive definiteness means that the inner product of a vector with itself is always positive.

How is the inner product related to orthogonality?

Orthogonality is a concept in mathematics that refers to two vectors being perpendicular to each other. The inner product of two orthogonal vectors is equal to 0, which means that they are perpendicular. This property is used in applications such as finding the shortest distance between a point and a line.

What are some real-world applications of the inner product?

The inner product has many applications in physics, engineering, and computer science. It is used in calculating work and energy in mechanics, finding the optimal direction in machine learning algorithms, and representing signals in signal processing. It is also used in geometry to calculate distances and angles between vectors.

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