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How are invariant lines/planes linked to eigenvalues/vectors? watch

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    Could someone explain the link between invariant lines / planes and eigenvalues/vectors?

    Is the eigenvalue the gradient of the invariant line y=mx under a given linear transformation? What about cases where the invariant line is y=mx + c?

    What does the eigenvector represent? The unit vector in the direction of the invariant line?

    Can finding the eigenvalues/vectors of a 3x3 matrix be used to find the equations of the planes which map to themselves under that matrix?
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    (Original post by SamKeene)
    Could someone explain the link between invariant lines / planes and eigenvalues/vectors?

    Is the eigenvalue the gradient of the invariant line y=mx under a given linear transformation? What about cases where the invariant line is y=mx + c?

    What does the eigenvector represent? The unit vector in the direction of the invariant line?

    Can finding the eigenvalues/vectors of a 3x3 matrix be used to find the equations of the planes which map to themselves under that matrix?
    eigenvectors

    are vectors therefore have "gradient" embodied in them
    e.g

    i + 2j represents y = 2x
    (if y = mx, then m is not the eigenvalue)

    they represent invariant directions in 2D and 3D space (or indeed in theoretical 4D+ space)

    in 3D space occasionally they represent invariant planes
    these is when the 3 equations do not reduce to the usual 2 but they are the same equation. ( I doubt if this is part of A level)
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    (Original post by TeeEm)
    eigenvectors

    are vectors therefore have "gradient" embodied in them
    e.g

    i + 2j represents y = 2x
    (if y = mx, then m is not the eigenvalue)

    they represent invariant directions in 2D and 3D space (or indeed in theoretical 4D+ space)

    in 3D space occasionally they represent invariant planes
    these is when the 3 equations do not reduce to the usual 2 but they are the same equation. ( I doubt if this is part of A level)
    So an eigenvector represents the lines which map onto themselves which pass through the origin.

    What about lines which map to themselves which do not pass through the origin? Or is that stepping outside of linear transformations?
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    (Original post by SamKeene)
    So an eigenvector represents the lines which map onto themselves which pass through the origin.

    What about lines which map to themselves which do not pass through the origin? Or is that stepping outside of linear transformations?
    these are not represented by eigenvectors
 
 
 
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