It's been a while since I took linear algebra, what does it mean for a matrix P to be positive-semidefinite?
penguin
I really like using these images to actually see the effect of the transformations
Max
@ngandhi It means $x^TPx \ge 0$ for all vectors $x$. Because $P$ is symmetric, we also know that all of its eigenvalues are real and non-negative.
emmurphy
What will the order reflect the outcome of the transform?
nouyang
I think the order does not matter when there are only rotation and scale. It will introduce problems when having translation.
bpopeck
I found this part a little confusing when thinking about how translation, scaling, rotation, and shearing make up our four fundamental transformations. If every transformation can be represented as a matrix and through singular value decomposition every matrix is a product of a rotation, axis-aligned scaling, and another rotation, then where do translation and shearing play a role? Can all translations and shears be expressed as a combination of rotations and scaling?
Max
@nouyang order matters if you have rotations - consider rotating 90 degrees about one axis and then another vs doing it in the opposite order. The only case in which order doesn't matter is pure scaling (i.e. diagonal matrices commute).
@bpopeck well, first remember that translation is not a linear transform - to represent "translations" we use shears in homogeneous coordinates. And a shear is just a non-uniform scaling about a particular axis, so it can be decomposed into rotations and scale (one of which can also include reflections). So really, rotations and scales are the most fundamental transforms.
It's been a while since I took linear algebra, what does it mean for a matrix P to be positive-semidefinite?
I really like using these images to actually see the effect of the transformations
@ngandhi It means $x^TPx \ge 0$ for all vectors $x$. Because $P$ is symmetric, we also know that all of its eigenvalues are real and non-negative.
What will the order reflect the outcome of the transform?
I think the order does not matter when there are only rotation and scale. It will introduce problems when having translation.
I found this part a little confusing when thinking about how translation, scaling, rotation, and shearing make up our four fundamental transformations. If every transformation can be represented as a matrix and through singular value decomposition every matrix is a product of a rotation, axis-aligned scaling, and another rotation, then where do translation and shearing play a role? Can all translations and shears be expressed as a combination of rotations and scaling?
@nouyang order matters if you have rotations - consider rotating 90 degrees about one axis and then another vs doing it in the opposite order. The only case in which order doesn't matter is pure scaling (i.e. diagonal matrices commute).
@bpopeck well, first remember that translation is not a linear transform - to represent "translations" we use shears in homogeneous coordinates. And a shear is just a non-uniform scaling about a particular axis, so it can be decomposed into rotations and scale (one of which can also include reflections). So really, rotations and scales are the most fundamental transforms.