How to solve a system of matrix equations?
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Given are two lists of 87 3x1 vectors each. I also know that when multiplying a 3x3 matrix by the nth vector of the first list, I get the nth vector of the second list. The 3x3 matrix is the same in all these calculations. So it's a system of equations with 87 equations and I want to solve for the content of the 3x3 matrix.
I've tried to approximate the solution with Machine Learning and Curve Fitting, but there are too few equations for that. How could I solve this problem mathematically?
matrices matrix-equations matrix-calculus
asked Dec 18 '18 at 13:21
Phoenix SmaugPhoenix Smaug
11
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add a comment |
$begingroup$
Given are two lists of 87 3x1 vectors each. I also know that when multiplying a 3x3 matrix by the nth vector of the first list, I get the nth vector of the second list. The 3x3 matrix is the same in all these calculations. So it's a system of equations with 87 equations and I want to solve for the content of the 3x3 matrix.
I've tried to approximate the solution with Machine Learning and Curve Fitting, but there are too few equations for that. How could I solve this problem mathematically?
matrices matrix-equations matrix-calculus
asked Dec 18 '18 at 13:21
Phoenix SmaugPhoenix Smaug
11
$endgroup$
$begingroup$
Are the vectors known?
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– Lucas Henrique
Dec 18 '18 at 13:25
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Yes, they are known.
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– Phoenix Smaug
Dec 18 '18 at 13:28
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$87$ data points is way enough in your case. Which techniques did you try ?
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– nicomezi
Dec 18 '18 at 13:30
1
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Each equation $M x_i = y_i$ gives us three linear equations involving the 9 unknown entries of the matrix $M$. So altogether you have $87 cdot 3 = 261$ linear equations in 9 unknown variables. You can find a least squares solution to this overdetermined linear system.
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– littleO
Dec 18 '18 at 13:33
add a comment |
$begingroup$
Given are two lists of 87 3x1 vectors each. I also know that when multiplying a 3x3 matrix by the nth vector of the first list, I get the nth vector of the second list. The 3x3 matrix is the same in all these calculations. So it's a system of equations with 87 equations and I want to solve for the content of the 3x3 matrix.
I've tried to approximate the solution with Machine Learning and Curve Fitting, but there are too few equations for that. How could I solve this problem mathematically?
matrices matrix-equations matrix-calculus
asked Dec 18 '18 at 13:21
Phoenix SmaugPhoenix Smaug
11
$endgroup$
Given are two lists of 87 3x1 vectors each. I also know that when multiplying a 3x3 matrix by the nth vector of the first list, I get the nth vector of the second list. The 3x3 matrix is the same in all these calculations. So it's a system of equations with 87 equations and I want to solve for the content of the 3x3 matrix.
I've tried to approximate the solution with Machine Learning and Curve Fitting, but there are too few equations for that. How could I solve this problem mathematically?
matrices matrix-equations matrix-calculus
matrices matrix-equations matrix-calculus
asked Dec 18 '18 at 13:21
Phoenix SmaugPhoenix Smaug
11
asked Dec 18 '18 at 13:21
Phoenix SmaugPhoenix Smaug
11
asked Dec 18 '18 at 13:21
Phoenix SmaugPhoenix Smaug
11
asked Dec 18 '18 at 13:21
Phoenix SmaugPhoenix Smaug
11
asked Dec 18 '18 at 13:21
Phoenix SmaugPhoenix Smaug
11
11
$begingroup$
Are the vectors known?
$endgroup$
– Lucas Henrique
Dec 18 '18 at 13:25
$begingroup$
Yes, they are known.
$endgroup$
– Phoenix Smaug
Dec 18 '18 at 13:28
$begingroup$
$87$ data points is way enough in your case. Which techniques did you try ?
$endgroup$
– nicomezi
Dec 18 '18 at 13:30
1
$begingroup$
Each equation $M x_i = y_i$ gives us three linear equations involving the 9 unknown entries of the matrix $M$. So altogether you have $87 cdot 3 = 261$ linear equations in 9 unknown variables. You can find a least squares solution to this overdetermined linear system.
$endgroup$
– littleO
Dec 18 '18 at 13:33
add a comment |
$begingroup$
Are the vectors known?
$endgroup$
– Lucas Henrique
Dec 18 '18 at 13:25
$begingroup$
Yes, they are known.
$endgroup$
– Phoenix Smaug
Dec 18 '18 at 13:28
$begingroup$
$87$ data points is way enough in your case. Which techniques did you try ?
$endgroup$
– nicomezi
Dec 18 '18 at 13:30
1
$begingroup$
Each equation $M x_i = y_i$ gives us three linear equations involving the 9 unknown entries of the matrix $M$. So altogether you have $87 cdot 3 = 261$ linear equations in 9 unknown variables. You can find a least squares solution to this overdetermined linear system.
$endgroup$
– littleO
Dec 18 '18 at 13:33
$begingroup$
Are the vectors known?
$endgroup$
– Lucas Henrique
Dec 18 '18 at 13:25
$begingroup$
Are the vectors known?
$endgroup$
– Lucas Henrique
Dec 18 '18 at 13:25
$begingroup$
Yes, they are known.
$endgroup$
– Phoenix Smaug
Dec 18 '18 at 13:28
$begingroup$
Yes, they are known.
$endgroup$
– Phoenix Smaug
Dec 18 '18 at 13:28
$begingroup$
$87$ data points is way enough in your case. Which techniques did you try ?
$endgroup$
– nicomezi
Dec 18 '18 at 13:30
$begingroup$
$87$ data points is way enough in your case. Which techniques did you try ?
$endgroup$
– nicomezi
Dec 18 '18 at 13:30
1
1
$begingroup$
Each equation $M x_i = y_i$ gives us three linear equations involving the 9 unknown entries of the matrix $M$. So altogether you have $87 cdot 3 = 261$ linear equations in 9 unknown variables. You can find a least squares solution to this overdetermined linear system.
$endgroup$
– littleO
Dec 18 '18 at 13:33
$begingroup$
Each equation $M x_i = y_i$ gives us three linear equations involving the 9 unknown entries of the matrix $M$. So altogether you have $87 cdot 3 = 261$ linear equations in 9 unknown variables. You can find a least squares solution to this overdetermined linear system.
$endgroup$
– littleO
Dec 18 '18 at 13:33
add a comment |
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$begingroup$
Are the vectors known?
$endgroup$
– Lucas Henrique
Dec 18 '18 at 13:25
$begingroup$
Yes, they are known.
$endgroup$
– Phoenix Smaug
Dec 18 '18 at 13:28
$begingroup$
$87$ data points is way enough in your case. Which techniques did you try ?
$endgroup$
– nicomezi
Dec 18 '18 at 13:30
1
$begingroup$
Each equation $M x_i = y_i$ gives us three linear equations involving the 9 unknown entries of the matrix $M$. So altogether you have $87 cdot 3 = 261$ linear equations in 9 unknown variables. You can find a least squares solution to this overdetermined linear system.
$endgroup$
– littleO
Dec 18 '18 at 13:33