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
 
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  • #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!
 

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

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.

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.

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.

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.

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.

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