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Ax=b, if matrix A is unknown?

  • Thread starter Genxi
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  • #1
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Homework Statement



I am trying to solve for the A matrix (3 x 3). I know matrix x is (3 x 1) and matrix b is (3 x 1), how do I go about solving for matrix A?


The Attempt at a Solution



I have not attempted this as I don't know the rules to initiate this problem.

Please offer me some hints or rules I should know on how to solve this problem


Thanks,


Genxi
 

Answers and Replies

  • #2
K29
108
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Well you could re-arrange it with matrix algebra:

[itex]Ax=b[/itex]

[itex]Axx^{T}=bx^{T}[/itex]

[itex]A=\frac{bx^{T}}{xx^{T}}[/itex]

Note the last step is allowed because [itex]xx^{T}[/itex] is a scalar.(You can't divide matrices by matrices)

Familiarise yourself with the transpose of a matrix (in this case, a column vector) and matrix multiplication, and perhaps rules of matrix algebra and that should be all you need to understand the above.

This link may be helpful http://people.hofstra.edu/stefan_waner/RealWorld/Summary3.html [Broken], though it may go into much more depth than you need.



.
 
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  • #3
DryRun
Gold Member
838
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Other than what K29 has suggested, here are some methods commonly used to solve systems of linear equations (matrix equations). You can do a little research, or refer in your book/s:

1. Cramer's rule.
2. Gauss elimination method (reduce to row echelon form)
3. LU-factorisation method
4. Using matrix inversion.

Personally, i prefer the Gauss elimination method which is quicker.
 
  • #4
Ray Vickson
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Well you could re-arrange it with matrix algebra:

[itex]Ax=b[/itex]

[itex]Axx^{T}=bx^{T}[/itex]

[itex]A=\frac{bx^{T}}{xx^{T}}[/itex]

Note the last step is allowed because [itex]xx^{T}[/itex] is a scalar.(You can't divide matrices by matrices)

Familiarise yourself with the transpose of a matrix (in this case, a column vector) and matrix multiplication, and perhaps rules of matrix algebra and that should be all you need to understand the above.

This link may be helpful http://people.hofstra.edu/stefan_waner/RealWorld/Summary3.html [Broken], though it may go into much more depth than you need.



.
This is incorrect. Since x and b are column vectors, the objects xxT and bxT are 3×3 matrices, not scalars.

Anyway, if x and b are known but A is unknown, the equations Ax = b give 3 equations in the 9 unknowns aij, so the system is underdetermined. Additional information or some type of optimization criterion would need to be incorporated in order to obtain a unique solution.

RGV
 
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  • #5
D H
Staff Emeritus
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Well you could re-arrange it with matrix algebra:

[itex]Ax=b[/itex]

[itex]Axx^{T}=bx^{T}[/itex]

[itex]A=\frac{bx^{T}}{xx^{T}}[/itex]

Note the last step is allowed because [itex]xx^{T}[/itex] is a scalar.(You can't divide matrices by matrices)
The last step is disallowed because [itex]xx^{T}[/itex] is a 3x3 matrix. What you might be able to do is post-multiply by the inverse of [itex]xx^{T}[/itex], but that too is disallowed because [itex]xx^{T}[/itex] is singular.

So let's go back to the start.


I am trying to solve for the A matrix (3 x 3). I know matrix x is (3 x 1) and matrix b is (3 x 1), how do I go about solving for matrix A?
You can't. Ax=b comprises three equations. However, you have nine unknowns, the nine elements of A. That's an underdetermined system. There are either no solutions or there are an infinite number of solutions.


Edit
I now see that Ray Vickson beat me to it.
 
  • #6
K29
108
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My mistake. I was working too quickly. Apologies to OP
 

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