How do you invert a 3D matrix? (Tensor inversion)

In summary, the conversation discusses solving a system of systems of equations involving a 3D tensor matrix, a vector, and a matrix. The speaker mentions not being able to find a clear explanation for inverting a tensor matrix, and asks if least-squares can be applied. They also mention converting a specific variable into a matrix for indexing purposes.
  • #1
BoltE
2
0
TL;DR Summary
I have three systems of equations in the form of Ax=b, where there are three different b-vectors, three different A-matrices, all of which use the same x-vector (A1x=b1, A2x=b2,A3x=b3). The goal is to solve for x. I can also write this as a tensor product: b_ij = sum_k (A_ijk x_k), where I would want to invert A to solve for X. I'm familar with regular linear systems where a is a 2D matrix and I could use a least-squares approach, MLEM, etc.
I would like to solve a system of systems of equations Ax=b where A is an n x m x p tensor (3D) matrix, x is a vector (n x 1), and b is a matrix (n x p). I haven't been able to find a clear walk-through of inverting a tensor like how one would invert a regular matrix to solve a system of linear equations. (or an iterative technique like MLEM).

Attached is a typed up version of the equations except with different variables where b = N, A = R, and x = S:

Figure.png
 
Physics news on Phys.org
  • #2
Why can't one just apply least-squares?

Okay, convert ##r_{i,j,k}## to a ##[n\times m p ]## matrix. It's just indexing.
 
  • #3
Whoops, I'll check this, thank you!
 

1. How do you define a 3D matrix?

A 3D matrix, also known as a tensor, is a mathematical object that stores data in a three-dimensional array. It is typically represented by a collection of numbers arranged in rows, columns, and layers.

2. Why would you need to invert a 3D matrix?

Inverting a 3D matrix allows for solving equations involving tensors, which are commonly used in fields such as physics, engineering, and computer graphics. It also helps with data manipulation and analysis in machine learning and data science.

3. What is the process for inverting a 3D matrix?

The process for inverting a 3D matrix involves finding the inverse of each individual 2D matrix within the tensor and then combining them to form the inverse of the entire tensor. This can be done using various mathematical techniques such as Gaussian elimination, LU decomposition, or singular value decomposition.

4. Are there any limitations or restrictions when inverting a 3D matrix?

Yes, there are certain restrictions and limitations when inverting a 3D matrix. The matrix must be square and non-singular, meaning it has a non-zero determinant. Additionally, the matrix must be invertible, which means it has a unique solution.

5. What are some common applications of inverting 3D matrices?

Inverting 3D matrices has various applications in different fields. It is commonly used in computer graphics for 3D transformations and in physics for solving systems of equations. It is also used in machine learning for data processing and analysis, such as in image and signal processing.

Similar threads

I Want to understand how to express the derivative as a matrix
    • Linear and Abstract Algebra
Replies
8
Views
1K
A How to augment a machine learning matrix?
    • Linear and Abstract Algebra
Replies
10
Views
145
I Linear least squares regression for model matrix identification
    • Linear and Abstract Algebra
Replies
5
Views
1K
B Is AB Invertible If n < m and B has a Non-Trivial Kernel?
    • Linear and Abstract Algebra
Replies
6
Views
1K
A Solutions of Matrix Equation A X B^T = C
    • Linear and Abstract Algebra
Replies
1
Views
2K
I General Method for Mapping an Ellipsoid to Unit Sphere
    • Linear and Abstract Algebra
Replies
1
Views
929
Vectors as geometric objects and vectors as any mathematical objects
    • Linear and Abstract Algebra
Replies
27
Views
1K
B Understanding Invertible Matrices and Homogenous Systems
    • Linear and Abstract Algebra
Replies
15
Views
1K
A Solve Badly Scaled Problem with Newton-Chord Technique
    • Linear and Abstract Algebra
Replies
12
Views
2K
I A regular matrix <=> mA isomorphism
    • Linear and Abstract Algebra
Replies
11
Views
2K

Hot Threads

Recent Insights

Back
Top