Any notation for component-by-component vector multiplication?

In summary: The matrix is called a projection matrix because it squares to itself: P^2 = P. It can be applied as many times as we like: P^{m+n} = P^m P^n. In particular, for a two-dimensional plane we can project twice onto the same direction, and get the same thing as projecting once:\frac{aa^T}{a^T a} \; \frac{aa^T}{a^T a} \; x = \frac{aa^T}{a^T a} \; x.So it's a sort of idempotent operation, like the identity matrix is.---In summary, there are various ways to express the operator needed to transform a vector and
  • #1
Curl
758
0
I have a scalar function and a vector function and I need to make a scalar function as so:

k=[kx ky kz]
T=T(x,y,z)

Function I want:

div(k ? gradT) where "?" would be some operator that multiplies each component of k and gradT to make the vector [kx ∂T/dx , ky ∂T/∂y , kz ∂T/∂z]

that way I can apply the divergence operator and get:
∂/∂x (kx ∂T/∂x) + ∂/∂y (ky ∂T/∂y) + ∂/∂z (kz ∂T/∂z)

So is there some way to express this using elementary notation?
 
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  • #2
Certeinly there is a notation of that operator, in general if you want to have a operator of that kind you have to put:

[tex]k\nabla\cdot(A)[/tex]

Where A is the vector field that you want to operate.

Greetings
 
  • #3
arsenal997 said:
Certeinly there is a notation of that operator, in general if you want to have a operator of that kind you have to put:

[tex]k\nabla\cdot(A)[/tex]

Where A is the vector field that you want to operate.

Greetings

k is a vector... what is [tex] \vec k \nabla [/tex] ??
 
  • #4
I believe it will be

[tex] \vec \nabla \cdot \left( ( \vec \nabla T)^T \cdot I_3 \vec k \right)^T [/tex]

I think this works
 
Last edited:
  • #5
Oh yes... the superscript T means transpose. Treat grad T as a 3 X 1 matrix, and k also as a 3 X 1 matrix. I3 is just the identity matrix for R3.
 
  • #6
Actually...

[tex] \left( ( \mathbf{ \nabla } T)^T \mathbf{ I_3 } \textbf{ k } \right) \mathbf{ \nabla } [/tex]

This may be a better way to put it. If we interpret nabla as a 3 X 1 matrix and k is also a 3 X 1 matrix
 
  • #7
I can't see what you are trying to do. The Identity matrix does nothing, and multiplying a 1x3 matrix by a 1x3 is not defined.
 
  • #8
Curl said:
I can't see what you are trying to do. The Identity matrix does nothing, and multiplying a 1x3 matrix by a 1x3 is not defined.

when you multiply k by the identity you get a diagonal matrix with the components of k along the diagonals. Then when you multiply the transpose of the gradient of T by this 3 X 3 matrix you get a 1 X 3 matrix inside of the parenthesis. Then the nabla matrix is 3 X 1 so we have a 1 X 3 multiplied by a 3 X 1 which gives the sum of the products of the components.
 
  • #9
AlexChandler said:
when you multiply k by the identity you get a diagonal matrix with the components of k along the diagonals. Then when you multiply the transpose of the gradient of T by this 3 X 3 matrix you get a 1 X 3 matrix inside of the parenthesis. Then the nabla matrix is 3 X 1 so we have a 1 X 3 multiplied by a 3 X 1 which gives the sum of the products of the components.

Haha I am sorry you are absolutely right. Let me think about this for a moment :biggrin:
 
  • #10
If you can find an operation that will transform k into a diagonal matrix, then the above function should work if you replace I3 k with that operation.
 
  • #11
You could express it with the dyadic product as

[tex]\nabla \cdot \text{diag}(\textbf{k} \otimes \nabla T) = \nabla \cdot \text{diag}([k] [\nabla T]^T),[/tex]

taking "diag" to mean "form a vector whose components are the diagonal entries of this matrix". Here [k] is a 3x1 matrix (a column vector), and the transpose of [del T] a 1x3 matrix (row vector), so that their product is a 3x3 matrix.

Or simply use the summation sign:

[tex]\sum_{i=1}^{n} \partial_i (k_i \partial_i T) = \sum_{i=1}^{n} \frac{\partial }{\partial x_i}\left ( k_i \frac{\partial T}{\partial x_i} \right ).[/tex]
 
  • #12
Rasalhague said:
You could express it with the dyadic product as

[tex]\nabla \cdot \text{diag}(\textbf{k} \otimes \nabla T) = \nabla \cdot \text{diag}([k] [\nabla T]^T),[/tex]

taking "diag" to mean "form a vector whose components are the diagonal entries of this matrix".

Nice! This is the kind of thing I was trying to find. What course can you take to cover these kinds of operations? A second semester of linear algebra?
 
  • #13
I don't know. I'm learning this stuff from books and the internet, and have never formally taken a course in linear algebra, so I don't know where and when such things would normally be taught. There's a section on dyadics in Snider & Davis: Vector Analysis, 6th edition. Gilbert Strang demonstrates the technique of making an n x n matrix out of vectors by multiplying a column vector by its transpose on the right in one of his MIT linear algebra lectures, which are online here

http://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/video-lectures/

Looking at the titles, it's probably Lecture 16: Projection matrices and least squares, or perhaps Lecture 15: Projections onto subspaces. Another place I've seen something similar is in presentations of the relativistic velocity-addition formula, and that did involve the identity matrix. I thought there was an example on the Wikipedia page: http://en.wikipedia.org/wiki/Velocity_addition But it looks like they've replaced it now; or maybe it was a different page where I saw it. I can post details if you're interested.

The projection matrix idea works like this. Suppose we want to project a vector x onto a vectot a, then we can this as a matrix equation:

[tex]\frac{a^Tx}{a^T a} \; a = \frac{aa^T}{a^T a} \; x = Px.[/tex]

(If a is unit length, we don't need to worry about the denominator.)
 

1. What is component-by-component vector multiplication?

Component-by-component vector multiplication is a mathematical operation where corresponding components of two vectors are multiplied together to form a new vector. This can also be referred to as element-wise multiplication.

2. How is component-by-component vector multiplication different from regular vector multiplication?

In regular vector multiplication, the two vectors are multiplied using the rules of matrix multiplication. This results in a single scalar value. In component-by-component vector multiplication, the two vectors are multiplied by multiplying their corresponding components, resulting in a new vector with the same dimension as the original vectors.

3. What is the notation used for component-by-component vector multiplication?

The most common notation for component-by-component vector multiplication is using the symbol "⊙". This is also known as the Hadamard product or the element-wise product.

4. How is component-by-component vector multiplication useful in science?

Component-by-component vector multiplication is useful in various fields of science, such as physics, engineering, and computer science. It can be used to perform calculations involving vectors, such as calculating the force or acceleration of an object in a specific direction.

5. Can component-by-component vector multiplication be applied to vectors of different dimensions?

No, component-by-component vector multiplication can only be performed on vectors with the same dimensions. If the vectors have different dimensions, they cannot be multiplied component-by-component.

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