What does a non-invertible affine transformation look like geometrically in terms of rotation/shear/scaling?

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For whatever reason, the scipy implementation of affine transformation requires an inverse transformation matrix: https://docs.scipy.org/doc/scipy/reference/generated/scipy.ndimage.affine_transform.html



How can I know in advance whether a certain combination of scaling/rotation/shearing will give me a non-invertible matrix, and what would that look like geometrically?




linear-algebra geometry linear-transformations




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    For whatever reason, the scipy implementation of affine transformation requires an inverse transformation matrix: https://docs.scipy.org/doc/scipy/reference/generated/scipy.ndimage.affine_transform.html



    How can I know in advance whether a certain combination of scaling/rotation/shearing will give me a non-invertible matrix, and what would that look like geometrically?




    linear-algebra geometry linear-transformations




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      For whatever reason, the scipy implementation of affine transformation requires an inverse transformation matrix: https://docs.scipy.org/doc/scipy/reference/generated/scipy.ndimage.affine_transform.html



      How can I know in advance whether a certain combination of scaling/rotation/shearing will give me a non-invertible matrix, and what would that look like geometrically?




      linear-algebra geometry linear-transformations




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      For whatever reason, the scipy implementation of affine transformation requires an inverse transformation matrix: https://docs.scipy.org/doc/scipy/reference/generated/scipy.ndimage.affine_transform.html



      How can I know in advance whether a certain combination of scaling/rotation/shearing will give me a non-invertible matrix, and what would that look like geometrically?





      linear-algebra geometry linear-transformations





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      asked Aug 7 at 18:29









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          A non-invertible transformation collapses the space along some direction(s). If you know about eigenvalues and eigenvectors, the eigenvectors of $0$ give you the directions in which this collapse happens.



          Rotations are isometries, so they never contribute to making a transformation non-invertible. A shear shouldn’t cause any collapse, either. It shifts things parallel to some direction, which keeps areas/volumes constant, so there’s no collapse there, either. That leaves scaling as the likely culprit, but that’s easy to detect: the result is not invertible iff any of the scale factors is zero.






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            A non-invertible transformation collapses the space along some direction(s). If you know about eigenvalues and eigenvectors, the eigenvectors of $0$ give you the directions in which this collapse happens.



            Rotations are isometries, so they never contribute to making a transformation non-invertible. A shear shouldn’t cause any collapse, either. It shifts things parallel to some direction, which keeps areas/volumes constant, so there’s no collapse there, either. That leaves scaling as the likely culprit, but that’s easy to detect: the result is not invertible iff any of the scale factors is zero.






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              up vote
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              A non-invertible transformation collapses the space along some direction(s). If you know about eigenvalues and eigenvectors, the eigenvectors of $0$ give you the directions in which this collapse happens.



              Rotations are isometries, so they never contribute to making a transformation non-invertible. A shear shouldn’t cause any collapse, either. It shifts things parallel to some direction, which keeps areas/volumes constant, so there’s no collapse there, either. That leaves scaling as the likely culprit, but that’s easy to detect: the result is not invertible iff any of the scale factors is zero.






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                up vote
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                up vote
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                down vote









                A non-invertible transformation collapses the space along some direction(s). If you know about eigenvalues and eigenvectors, the eigenvectors of $0$ give you the directions in which this collapse happens.



                Rotations are isometries, so they never contribute to making a transformation non-invertible. A shear shouldn’t cause any collapse, either. It shifts things parallel to some direction, which keeps areas/volumes constant, so there’s no collapse there, either. That leaves scaling as the likely culprit, but that’s easy to detect: the result is not invertible iff any of the scale factors is zero.






                share|cite|improve this answer













                A non-invertible transformation collapses the space along some direction(s). If you know about eigenvalues and eigenvectors, the eigenvectors of $0$ give you the directions in which this collapse happens.



                Rotations are isometries, so they never contribute to making a transformation non-invertible. A shear shouldn’t cause any collapse, either. It shifts things parallel to some direction, which keeps areas/volumes constant, so there’s no collapse there, either. That leaves scaling as the likely culprit, but that’s easy to detect: the result is not invertible iff any of the scale factors is zero.







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                answered Aug 7 at 20:16









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