# Need help in Matrix Decomposition

• Jan 20th 2010, 10:40 AM
dimper129
Need help in Matrix Decomposition
Hi all,

I have the product of a 4x3 rectangular matrix and its transpose (the transpose is a 3x4 matrix). The product is a 4x4 symmetric matrix.

How can I find the original 4x3 rectangular matrix, if I know the values of all the elements of the 4x4 product matrix?

If anyone could help me solve this problem, i would realy appreciate it.
Thanks
• Jan 20th 2010, 11:52 AM
pickslides
Quote:

Originally Posted by dimper129
Hi all,

I have the product of a 4x3 rectangular matrix and its transpose (the transpose is a 3x4 matrix). The product is a 4x4 symmetric matrix.

How can I find the original 4x3 rectangular matrix, if I know the values of all the elements of the 4x4 product matrix?

If anyone could help me solve this problem, i would realy appreciate it.
Thanks

The only matrix you have is the 4x4? Can you post this matrix and any others?
• Jan 21st 2010, 02:41 AM
dimper129
Quote:

Originally Posted by pickslides
The only matrix you have is the 4x4? Can you post this matrix and any others?

Yes the only matrix I have is the 4x4 product.
e.g. if

$\displaystyle \left[\begin{matrix} a_1 & a_2 & a_3 \\ a_4 & a_5 & a_6 \\ a_7 & a_8 & a_9 \\ a_{10} & a_{11} & a_{12} \end{matrix}\right]\left[\begin{matrix} a_1 & a_4 & a_7 & a_{10} \\ a_2 & a_5 & a_8 & a_{11} \\ a_3 & a_6 & a_9 & a_{12} \end{matrix}\right] = \left[\begin{matrix} 0.0018 & -0.0062 & -0.0009 & 0.0021 \\ -0.0062 & 0.0537 & -0.0027 & 0.0014 \\ -0.0009 & -0.0027 & 0.1142 & 0.0063 \\ 0.0021 & 0.0014 & 0.0063 & 0.1005 \end{matrix}\right]$

then how can I find the values of $\displaystyle a_1, a_2 ...$ upto $\displaystyle a_{12}$

Thanks
• Jan 24th 2010, 02:28 AM
dimper129
It makes 10 equations in 12 unkowns like

$\displaystyle {a_1}^2 + {a_2}^2 + {a_1}^2 = 0.0018$
$\displaystyle a_1a_4 + a_2a_5 + a_3a_6 = -0.0062$
$\displaystyle a_1a_7 + a_2a_8 + a_3a_9 = -0.0009$
$\displaystyle a_1a_{10} + a_2a_{11} + a_3a_{12} = 0.0012$
$\displaystyle {a_4}^2 + {a_5}^2 + {a_6}^2 = 0.0537$
$\displaystyle a_4a_{7} + a_5a_{7} + a_6a_{9} = -0.0027$
$\displaystyle a_4a_{10} + a_5a_{11} + a_8a_{12} = 0.0014$
$\displaystyle {a_7}^2 + {a_8}^2 + {a_9}^2 = 0.1142$
$\displaystyle a_7a_{10} + a_8a_{11} + a_8a_{12} = 0.0063$
$\displaystyle {a_{10}}^2 + {a_{11}}^2 + {a_{12}}^2 = 0.1005$

we can solve (approximate) this non-linear system of equations in MATLAB using "fsolve" function. But this does not solve my problem because I need exact solution.

Does anyone know how to do it?

Thanks