# Thread: Need help in Matrix Decomposition

1. ## 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

2. 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?

3. 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

4. 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