# Minimization of matrix of vectors in polar field

The problem I am facing is the reduction of vibrations of a rotating object. I have a series of vibration measurements taken at 5 different states with magnitude and phase components, and a set of coefficients corresponding to 4 adjustment types and each state. I'm trying to find the adjustments necessary to minimize the resulting vibration.

Basically, it's just a system of equations problem that I would like an automated solution for. For a single vibration and adjustment, the equation would be $\vec{v}_f=\vec{v}_i+a\vec{c}$, where $a$ is the scalar adjustment. I break the components into Cartesian coordinates from their polar form and solve the system $$\left\{\begin{array}{1} v_{x_f}=v_{x_i}+aC_x\\ v_{y_f}=v_{y_i}+aC_y\\ \end{array}\right.$$

solving for $a$ such that $\sqrt{v_{x_f}^2+v_{y_f}^2}$ is minimized.

I've set up the vibrations, adjustments and coefficients into matrices, but I'm still a little foggy on what operations I need to perform to solve it. $$\vec{V}_f = \begin{bmatrix} \vec{v}_{1_f} \\ \vec{v}_{2_f} \\ \vec{v}_{3_f} \\ \vec{v}_{4_f} \\ \vec{v}_{5_f} \\ \end{bmatrix} , \vec{V}_i = \begin{bmatrix} \vec{v}_{1_i} \\ \vec{v}_{2_i} \\ \vec{v}_{3_i} \\ \vec{v}_{4_i} \\ \vec{v}_{5_i} \\ \end{bmatrix} , \vec{C} = \begin{bmatrix} \vec{c}_{(1,a)} & \vec{c}_{(1,b)} & \vec{c}_{(1,c)} & \vec{c}_{(1,d)} \\ \vec{c}_{(2,a)} & \vec{c}_{(2,b)} & \vec{c}_{(2,c)} & \vec{c}_{(2,d)} \\ \vec{c}_{(3,a)} & \vec{c}_{(3,b)} & \vec{c}_{(3,c)} & \vec{c}_{(3,d)} \\ \vec{c}_{(4,a)} & \vec{c}_{(4,b)} & \vec{c}_{(4,c)} & \vec{c}_{(4,d)} \\ \vec{c}_{(5,a)} & \vec{c}_{(5,b)} & \vec{c}_{(5,c)} & \vec{c}_{(5,d)} \\ \end{bmatrix} , A = \begin{bmatrix} a_{a}\\ a_{b}\\ a_{c}\\ a_{d}\\ \end{bmatrix}$$ $$\vec{V}_{f} = \vec{V}_i + \vec{C} A$$ Essentially I believe I need to find the values for $a$ such that the magnitudes of the vibrations are minimized. Is it possible to even solve using this equation using matrices, or am I completely off base?

I would really appreciate any discussion that might point me on the right track.

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What's $A$, and where did $\vec{M}_f$ go in your final equation? –  Ｊ. Ｍ. May 6 '11 at 18:11
I made a change halfway through and forgot to change everything back to A..."Adjustment" sounded better than "Move". The most recent edit reflects the changes to 'A' from 'M'. –  TriNitroToluene May 6 '11 at 18:16
The dimensions of $\vec A$ and $\vec C$ don't seem to conform; you've four columns in $\vec A$ and five rows in $\vec C$... –  Ｊ. Ｍ. May 6 '11 at 18:24
On the other hand, if you meant $C\vec{A}$, then least squares is what'll save your bacon. –  Ｊ. Ｍ. May 6 '11 at 18:25
Basically, what I'm doing with the coefficient matrix is showing how much the vibration point will move at the particular state if adjusted by $a$. –  TriNitroToluene May 6 '11 at 18:50