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## Main Question or Discussion Point

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?

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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

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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

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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.

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?

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