Rotation that maximises a vector length

Hi all,

I have a problem that is more related to linear algebra than Matlab itself, even if I would like to solve it with Matlab.

I have a 3*3 tensor 'K' that applied to a vector 'b' returns another vector 'm'

m = K*b;

Let's suppose that the vector 'b' has a fixed magnitude but the orientation is arbitrary. Now, I would like to find the rotation to apply to 'b' such that the magnitude of 'm' is maximum.

For example (but this is trivial), let's assume unitary 'b' and

K = [2 0 0; 0 0 0; 0 0 0];

The direction with maximum 'm' is with

bR = [1 0 0];

The solution can be not unique, and the problem is not well posed, but I wonder if there is a procedure to extract at least one direction of 'b' that maximises the length of 'm' for a generic tensor 'M'.

Thank you in advance Fabio

[Torsten]

If I remember correctly, b is eigenvector to the largest eigenvalue of the (symmetric positive semidefinite) matrix K'*K.

Maybe somebody of the newsgroup can confirm.

Best wishes

Torsten.

[Fabio Freschi]

Thank you! With a simple verification it looks like it works (see below). Torsten, could you suggest where to find out some help on this issue? I don't know what to search for on google.

% tensor
K = rand(3)*eye(3);

% number of points
n = 1000;
% b1 and b2 components
[b1,b2] = meshgrid(linspace(0,1,n),linspace(0,1,n));
% b3 component
b3 = sqrt(1-b1.^2-b2.^2);
% identify the infeasible points
idx = b1.^2+b2.^2 > 1;

% calculate m = K*b
m = K*[b1(:) b2(:) b3(:)]';

% magnitude
mMod = sqrt(sum(m.^2,1));
% remove infeasible points
mMod(idx(:)) = NaN;
% reshape
mMod = reshape(real(mMod),size(b1));

% plot
h = figure;
surf(b1,b2,mMod,'edgeColor','none');

% find maximum and position
[mMax,iLinMax] = max(mMod(:));

% plot
figure(h), hold on
plot3(b1(iLinMax),b2(iLinMax),mMax,'bo');

% analytic solution
[V,D] = eig(K'*K,'vector');
% find position of max eigenvalue
[~,iEigMax] = max(D);
% get eigenvector and normalize
Vmax = V(:,iEigMax)/sqrt(sum(V(:,iEigMax).^2));
% calculate max m
mMaxEig = K*Vmax(:);

% plot
plot3(Vmax(1),Vmax(2),sqrt(sum(mMaxEig.^2)),'r*');

[Torsten]

Take a look at the chapter "Constrained Optimization" under

http://biocomp.cnb.csic.es/~coss/Docencia/algebra/tema8.pdf

You want to maximize

||Kx||^2 = (Kx)'(Kx) = x'K'Kx

under the constraint

||x||^2=x'*x=1

Best wishes

Torsten.

[Bjorn Gustavsson]

Torsten's suggestion seems correct, but that example K is not a rotation matrix since K*K.' is not an identity matrix.

[Torsten]

That's how I understand what the OP's questions:

K is a given matrix (it does not need to be a rotation matrix) together with an arbitrary vector b.

Now the OP searches for a rotation matrix R such that K*R*b has maximum length.

In other words:

The OP searches for a vector y with |y| = |b| such that |K*y| is maximized. And without loss of generality, we can assum |y| = 1.

Best wishes

Torsten.

[Fabio Freschi]

Torsten is right. K does not need to be a rotation matrix.

Thanks for your help!

Fabio