# Matrix Factorization: Spherical & Cartesian Vectors

• psholtz
In summary, the matrix T represents the relation between spherical and cartesian unit vectors and cannot be factored into simpler matrices. However, it can be decomposed using rotations and trigonometry identities to transform between different coordinate systems.
psholtz
The matrix giving the relation between spherical (unit) vectors and cartesian (unit) vectors can be expressed as:

$$\left( \begin{array}{c} \hat{r} \\ \hat{\phi} \\ \hat{\theta} \end{array} \right) = \left( \begin{array}{ccc} \sin\theta \cos\phi & \sin\theta \sin\phi & \cos\theta \\ -\sin\phi & \cos \phi & 0 \\ \cos\theta \cos\phi & \cos\theta \sin\phi & -\sin\theta \end{array}\right) \cdot \left( \begin{array}{c} \hat{x} \\ \hat{y} \\ \hat{z} \end{array} \right)$$

or

$$T = \left( \begin{array}{ccc} \sin\theta \cos\phi & \sin\theta \sin\phi & \cos\theta \\ -\sin\phi & \cos \phi & 0 \\ \cos\theta \cos\phi & \cos\theta \sin\phi & -\sin\theta \end{array}\right)$$

where phi is the polar angle and theta is the azimuthal angle.

Can this matrix T be factored into simpler matrices?

Hi psholtz!

But the short answer would be: no, you can't simplify this matrix.

There is more than one way to decompose this. Usually for this kind of thing you simply perform rotations to construct the transformation from one set of coordinates to the other. In your case it is a little weird since you didn't order your output coordinates in a right-handed way. By the way, I am assuming here that you mis-typed: phi is azimuthal and theta is polar?

Anyway, here is one option

$$T = \left( \begin{array}{ccc} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array}\right) \left( \begin{array}{ccc} \cos \theta & 0 & -\sin \theta \\ 0 & 1 & 0 \\ \sin \theta & 0 & \cos\theta \end{array}\right) \left( \begin{array}{ccc} \cos \phi & \sin \phi & 0 \\ -\sin \phi & \cos \phi & 0 \\ 0 & 0 & 1 \end{array} \right)$$

edit: The left-most matrix is not a rotation; a non-rotation is required here since your output coordinates are not in rigt-handed order. I mapped to (theta,phi,r), which is right-handed, and then used the left matrix to map to your ordering. Given my ordering, the right-most matrix rotates about z to make y and phi coincide, the middle matrix then rotates about the y-axis to make z and r coincide and x and theta coincide.

jason

Last edited:
The only way to solve this matrix is to try and use LU factorisation ,but if look @ entries (t11 and t13 are will never be reduced and this is similar to the other entries in the matrix ,HENCE this matrix can not be reduced to row echelon form. 4 an advice try and use trigonometry identites and see if u r goinn to find U. or u can try and use (sin) as the scalar multiple of T.

The only way to solve this matrix is to try and use LU factorisation ,but if look @ entries (t11 and t13 are will never be reduced and this is similar to the other entries in the matrix ,HENCE this matrix can not be reduced to row echelon form. 4 an advice try and use trigonometry identites and see if u r goinn to find U. or u can try and use (sin) as the scalar multiple of T.

## 1. What is matrix factorization?

Matrix factorization is a mathematical technique used to decompose a matrix into two or more matrices, with the goal of simplifying the representation of the original matrix and making it easier to analyze or manipulate.

## 2. What are spherical and cartesian vectors?

Spherical and cartesian vectors are two different coordinate systems used to represent points or vectors in three-dimensional space. In spherical coordinates, a vector is described by its distance from the origin, the angle it makes with the positive z-axis, and the angle it makes with the positive x-axis. In cartesian coordinates, a vector is described by its x, y, and z coordinates.

## 3. What is the purpose of using matrix factorization with spherical and cartesian vectors?

The purpose of using matrix factorization with spherical and cartesian vectors is to simplify the representation of vectors in three-dimensional space. By decomposing a matrix into two or more matrices, we can manipulate and analyze the vectors more easily, and also understand the relationships between them more intuitively.

## 4. How does matrix factorization work with spherical and cartesian vectors?

In matrix factorization, the original matrix is decomposed into two or more matrices using mathematical operations such as multiplication, addition, or subtraction. In the case of spherical and cartesian vectors, the original matrix represents the vectors in one coordinate system and the resulting matrices represent the vectors in the other coordinate system. This allows us to switch between the two systems and perform calculations more easily.

## 5. What are some real-world applications of matrix factorization with spherical and cartesian vectors?

Matrix factorization with spherical and cartesian vectors has various applications in fields such as computer graphics, physics, and engineering. For example, it can be used to simplify and speed up calculations in 3D modeling and animation, or to analyze and manipulate data in physics experiments. It can also be used in machine learning algorithms to reduce the dimensionality of data and improve performance.

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