Webpage title: How to Solve a 3D Matrix Equation

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    3d Matrix
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Discussion Overview

The discussion revolves around solving a 3D matrix equation of the form ai = bijkcjk, focusing on the challenge of determining the values of c given known values of a and b. Participants explore the implications of the dimensions involved and the nature of the system of equations, considering both theoretical and practical approaches to the problem.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant expresses uncertainty about solving for individual values of c, suggesting that there may be more unknowns than equations if the indices run from 1 to 3.
  • Another participant proposes that if n = m^2, the equations could be consistent, potentially allowing for a solution.
  • A participant identifies the system as underdetermined, emphasizing the need for the number of equations to match the number of unknowns for a solution to exist.
  • Further clarification is provided that with n set to m^2, there would be m^2 equations corresponding to m^2 unknowns, which could facilitate a solution.
  • One participant suggests "unwrapping" the indices to convert the matrix equation into a simpler form, allowing for standard linear algebra techniques to be applied.
  • The method of mapping indices to a single dimension is discussed, with a formula provided for transforming the indices into a column vector.

Areas of Agreement / Disagreement

Participants express differing views on the solvability of the equation based on the number of equations relative to the number of unknowns. While some agree that setting n = m^2 could lead to a solvable system, others remain skeptical about the feasibility of determining individual values of c.

Contextual Notes

Participants note the importance of maintaining the structure of the equations to avoid losing valuable information, which complicates the process of finding a solution. The discussion also highlights the potential complexity of the system depending on the definitions and constraints applied.

mikeph
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Hi,

I'm pretty rusty with solving linear equations, my equations are:

ai = bijkcjk

i = 1,... n
j, k = 1,... m

Would like to know c, given a and b. Need somewhere to start, without having to cover 3 years of notes that I took 5 years ago, thanks if anyone can point me in the right direction!

Comments: I know I could compact j and k into a single dimension and reduce it to a "simple" matrix equation but this process discards valuable information that I need later on. Or at least makes it difficult to retrieve this information, so I'd like to explore a way of solving the equations outright. Thanks.
 
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I'm not sure that you can solve for the individual values of c. It looks to me like you will have more uknown values than equations. If the indices run from 1 to 3, c will have 9 values but because of the contraction on both i and k, you will have only three equations.
 
Good point, I forgot to mention that. Let's say n = m^2, and the equations are consistent.
 
Yep - I think HallsofIvey is correct. The term we could use here is an underdetermined system.
 
HallsofIvy said:
I'm not sure that you can solve for the individual values of c. It looks to me like you will have more uknown values than equations. If the indices run from 1 to 3, c will have 9 values but because of the contraction on both i and k, you will have only three equations.

Yeah for a system of equations (unless those are complex nums, which I thought on first glance) you need the num of differing equations= the num of variables.
 
Ok, that condition is satisfied by setting n=m^2, now i,j = 1,... m, k = 1,... m^2 so we have m^2 equations in m^2 unknowns.
 
If n = m^2, then this is simple. First you need to "unwrap" c_jk into a single column vector c_l, where l now runs from 1 to m^2. b_ijk must be similarly unwrapped into b_il. In both cases, this is easily accomplished by defining

l = 3(j-1) + (k-1) + 1 = 3j + k - 3

l now runs from 1 to 9 as j and k run from 1 to 3, with each value of j and k being mapped to a unique l. Notice all I've done here is expand l in base 3. For general m^2, you would have

l = m(j - 1) + (k - 1) + 1 = mj + k - m

Once you have re-written your equation as

a_i = b_il c_l

it is easy to solve by standard linear algebra.

Finally, since each (j,k) pair maps to a unique l, it is also easy to get back c_jk the way you want it.
 

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