Webpage title: How to Solve a 3D Matrix Equation

  • Thread starter mikeph
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    3d Matrix
In summary, you would need to expand l in base 3 to get c_jk back. Yep - I think HallsofIvey is correct.
  • #1
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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  • #2
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.
 
  • #3
Good point, I forgot to mention that. Let's say n = m^2, and the equations are consistent.
 
  • #4
Yep - I think HallsofIvey is correct. The term we could use here is an underdetermined system.
 
  • #5
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.
 
  • #6
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.
 
  • #7
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.
 

What is a 3D matrix equation?

A 3D matrix equation is an equation that involves three-dimensional matrices. These matrices have three dimensions: rows, columns, and depth. In other words, they are a collection of numbers arranged in a three-dimensional grid.

Why do we need to solve 3D matrix equations?

3D matrix equations are used in various fields of science and engineering to model and solve complex problems. They are especially useful in computer graphics, physics, and economics.

How do you solve a 3D matrix equation?

To solve a 3D matrix equation, you can use various methods such as Gaussian elimination, Cramer's rule, or matrix inversion. The method used depends on the size and complexity of the equation.

What are the applications of solving 3D matrix equations?

Solving 3D matrix equations has many practical applications, including image and signal processing, computer animation, robotics, and optimization problems.

What are some challenges in solving 3D matrix equations?

One of the main challenges in solving 3D matrix equations is the computational complexity, especially for large matrices. Additionally, finding an appropriate method to solve the equation and handling errors and inaccuracies in the data can also be challenging.

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