How to Determine Current Directions from an Electrical Conductivity Tensor?

[jsmith12]

If I'm given the components of an electrical conductivity tensor, how can I work out the directions in which the current is biggest and the directions in which no current flows?

 

[HallsofIvy]

Find a coordinate system in which the tensor can be written as a diagonal matrix. (Writing the original tensor as a matrix, your coordinate axes will be the eigenvectors of the original matrix, the flow along those axes the eigenvalues.) The flow will be greatest in the direction of the eigenvector corresponding to the largest eigenvalue, 0 in any eigenspace corresponding to 0 eigenvalue.

 

[jsmith12]

Ok thanks. But if I'm trying to work this out for the matrix

2 -1 -1
-1 2 -1
-1 -1 2

which has eigenvalues 0, 3, 3.
The 0 eigenvalue gives (1,1,1) so that's the direction of no flow, but with eigenvalue 3 you can get lots of different eigenvectors so is the flow greatest in all of those directions?

 

[HallsofIvy]

Ok thanks. But if I'm trying to work this out for the matrix

2 -1 -1
-1 2 -1
-1 -1 2

which has eigenvalues 0, 3, 3.
The 0 eigenvalue gives (1,1,1) so that's the direction of no flow, but with eigenvalue 3 you can get lots of different eigenvectors so is the flow greatest in all of those directions?

The only eigenvectors corresponding to eigenvalue 3 all satisfy x+ y+ z= 0. That means that there is a 2 dimensional "eigenspace" of covering the plane x+ y+ z= 0. The flow is the same in all directions on that plane.

 

[Henok]

if the electric conductivity tensor is given how can i find the electric field with its related components