I have an equation of the form t=X^T D^-1 X (1), where X is a 3D vector X=[X Y Z] and D^-1 is the inverse of a 3x3 diffusion tensor D=[Dxx Dxy Dxz; Dxy Dyy Dyz; Dxz Dyz Dzz]. X^T means that the vector is transposed.
The 3D vector of X is the 3D distance for a set of points from the origin and t is the time for a pulse to get there.
Assuming that one of the eigenvectors is vertical (Dxz and Dyz are zero), equation (1) becomes
t=Dxx^-1 X^2+Dyy^-1 Y^2+Dzz^-1 Z^2+2 Dxy^-1 X Y (2)
The problem is to find the eigenvectors of D-1.
The principal horizontal axes of D will be those where the non-diagonal term Dxy approaches zero. The principal horizontal axes (x, y) are derived from the geographical system (xg, yg) by a rotation of θ about the z axis (θ is expressed counterclockwise in degrees between x axis and xg axis).
Knowing t and X for a set of points, a linear regression can be applied to equation (2).
My problem is that I assume that a multiple linear regression will be efficient, but I don't know how the angle θ is expressed in equation (2) so as to perform the multiple linear regression for each angle θ and see where Dxy approaches zero.
I would be grateful if anyone can help me with this.
Re: Eigenvector problem
Your questino is a bit confused. Do you want to perform the eigen decomposition of D^-1 ?
If this is the case, for 3x3 matrix you can even derive analytical solutions. You don't even need to invert D. You compute the roots determinant of | D - l*I |, where l is an eigen value and I is the identity matrix.
The 3 roots are the eigenvalues, and then you can compute the eigenvectors as :
U_i = [ d_xy * d_yz - d_xz * (d_yy - l_i) ]
[ d_xy * d_xz - d_yz * (d_xx - l_i) ]
[(d_xx - l_i) * (d_yy - l_i) - d_xy^2 ]
D and D^-1 share the same eigenvectors and their eigenvalues are just invert of each other.
Re: Eigenvector problem
Many thanks for your reply. Yes, the problem is a bit confused, as it confuses me as well. Although it looks like eigen decomposition problem, the thing is that I don't know
anything about the values of D. I have the values of the 3D vector X and for every X the value of t. As the vector X have the values of the coordinates of points in 3D space,
D is the tensor. So the problem is to see for which eigenvalues, D is diagonalized. This is supposed to be achieved through the linear regression of equation (2) for different angles of θ (0-180 degrees). When D is diagonalized, Dxy will be zero. Another way will be that assuming that the system is in the principal axis of the tensor D, then Dxy=0. Then again a linear regression can be applied to equation (2) with Dxy=0 for different θ. The principal axis and the eigenvalues of D will be those when the regression coefficient is maximum.
Again my problem is that I don't know how to write equation (2) for different θ in order to perform the linear regression.