4 ms·
Take a linear map from some space to itself, and ask: What lines (through the origin) are mapped back to themselves? Those are the eigenvectors, and the amount
by FabHK 4y ago
Take a linear map from some space to itself, and ask:
What lines (through the origin) are mapped back to themselves? Those are the eigenvectors, and the amount by which they're elongated or shortened are the eigenvalues.
So, if we talk about 3d space, and we rotate things - the rotation axis is unchanged. That's an eigenvector (with eigenvalue 1).
If we mirror things - any vector in the mirror plane remains unchanged, that's an eigenvector (with eigenvalue 1), the vector perpendicular to the mirror is unchanged, but flipped, so that's an eigenvector (with eigenvalue -1).
If we dilate everything along the x axis by a factor of 2, say, then the x axis is an eigenvector (with eigenvalue 2), while the y and z axis and any vector in that plane is an eigenvector (with eigenvalue 1). Any other vector is "tilted", so not mapped to itself, so not an eigenvector.
- MichaelZuo 4y agoWhat does 'through the origin' mean in a physical system?
- theGnuMe 4y agoCenter of mass; the object itself.
- pbhjpbhj 4y agoIt means it doesn't matter where it is: you can choose the origin, ie the point you measure from, it is arbitrary. Or another way of saying that is you can move the system to a different set of coordinates and it works in the same way. ... which means it's probably an imaginary physical system. Maybe a good physical example is a piece of cloth that warps in 2D, and shrinks, when washed? Eigenvectors would describe the warping (skew, say) and eigenvalues the shrinkage relative to the original warp and weft. Steve Brunton on YouTube has really good videos on eigenvectors & eigenvalues in context of matrix algebra (and then applied to simultaneous differential equations); https://youtube.com/watch?v=ZSGrJBS_qtc https://youtube.com/watch?v=ZSGrJBS_qtc .
- MichaelZuo 4y agoOkay, so that explains 'the origin'. Does 'through the origin' imply motion through 'the origin'?
- DennisP 4y agoWhen you're rotating something, the axis of rotation. That's the point that doesn't change in rotation ("maps to itself").
- mdup 4y agoIt means the eigenvalues will only give you information about the system relatively to the center of that system. Before describing any system, it's up to you (your "convention") to assert where is the zero-point of your world and in which directions the axes (x,y,z) are pointing. For instance, in the real world you can choose your 3D coordinate system such that your mirror, as a physical system, keeps the origin untouched (0,0,0) -> (0,0,0). If you decide the origin is a point on the mirror, the equations will be linear: mirror(X) = AX. However if you setup the origin some point far from the mirror, like the center of your eyes, the equations are no longer linear, but affine: mirror(X) = AX+B. Looking at the values of the "AX" part of the system would reveal you the mirroring plane, but now shifted by an offset of "+B" -- the distance between the mirror and your eyes -- because your choice of coordinates was not leaving the origin intact.