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FP4 geometrical interpretation of eigenvectors Watch

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    Hi

    I was working with eigenvectors of M representing invariant lines of transformations with matrix M.

    Am pastpapering, have come across the interpretation being a plane of invariant points. Please could someone explain to me when, and why, this happens? Is it possibly linked to there being a repeated eigenvalue?

    Cheers
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    (Original post by Vernish)
    Hi

    I was working with eigenvectors of M representing invariant lines of transformations with matrix M.

    Am pastpapering, have come across the interpretation being a plane of invariant points. Please could someone explain to me when, and why, this happens? Is it possibly linked to there being a repeated eigenvalue?

    Cheers
    Anyone please?
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    Eigenvectors indicate the direction which remain invariant and the eigenvalue determines the length.

    If the eigenvalue is 1 it means that the points on the invariant line are not stretched so you will have a line of invariant points.
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    (Original post by SherlockHolmes)
    Eigenvectors indicate the direction which remain invariant and the eigenvalue determines the length.

    If the eigenvalue is 1 it means that the points on the invariant line are not stretched so you will have a line of invariant points.
    Thanks for your reply! So how do planes come in? :confused:
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    (Original post by Vernish)
    Hi

    I was working with eigenvectors of M representing invariant lines of transformations with matrix M.

    Am pastpapering, have come across the interpretation being a plane of invariant points. Please could someone explain to me when, and why, this happens? Is it possibly linked to there being a repeated eigenvalue?

    Cheers
    If M is the identity matrix then every point of the plane is invariant. If M is a multiple of the identity matrix then every point of the plane will be mapped onto a point of the plane
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    (Original post by SherlockHolmes)
    Eigenvectors indicate the direction which remain invariant and the eigenvalue determines the length.

    If the eigenvalue is 1 it means that the points on the invariant line are not stretched so you will have a line of invariant points.
    Hi Mr Holmes, I was looking at a question about the eigenvectors of a matrix and was a bit confused about what this actually meant. Could you clarify this for me please?
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    (Original post by pleasedtobeatyou)
    Hi Mr Holmes, I was looking at a question about the eigenvectors of a matrix and was a bit confused about what this actually meant. Could you clarify this for me please?
    An eigenvector A, must obey the following equation:

    Av=\lambda v

    where v is a vector and a \lambda is a multiple of v.
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    (Original post by SherlockHolmes)
    An eigenvector A, must obey the following equation:

    Av=\lambda v

    where v is a vector and a \lambda is a multiple of v.
    Thank you Mr Holmes, you help has been invaluable.

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    (Original post by Vernish)
    Hi

    I was working with eigenvectors of M representing invariant lines of transformations with matrix M.

    Am pastpapering, have come across the interpretation being a plane of invariant points. Please could someone explain to me when, and why, this happens? Is it possibly linked to there being a repeated eigenvalue?

    Cheers
    I'm assuming that this is for a 3x3 matrix (although it works for any higher dimension as well), call is M

    If you have any 2 eigenvectors (say u,v with eigenvalues p,q), these span a plane in \mathbb{R}^3. Any point in this plane can be written in the form au+bv for some real numbers a,b. So if w is a point in the plane,

    Mw=M(au+bv)=apu+bqv which is a point in the plane (since it is a linear combination of u,v).
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    (Original post by james22)
    I'm assuming that this is for a 3x3 matrix (although it works for any higher dimension as well), call is M

    If you have any 2 eigenvectors (say u,v with eigenvalues p,q), these span a plane in \mathbb{R}^3. Any point in this plane can be written in the form au+bv for some real numbers a,b. So if w is a point in the plane,

    Mw=M(au+bv)=apu+bqv which is a point in the plane (since it is a linear combination of u,v).
    Thank you, I think that's made sense.
 
 
 
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