I cannot visualise an oblique projection. I understood the orthogonal one:(adsbygoogle = window.adsbygoogle || []).push({});

The orthogonal projection is P=U[tex]\cdot[/tex]U^{*}, where U is an orthonormal matrix (basis of a subspace) : U^{*}[tex]\cdot[/tex]U=I .

Now the projection of matrix A on U vectors is: P_{A}=U^{*}[tex]\cdot[/tex]A[tex]\cdot[/tex]U .

For the orthogonal projection, for a vector case, the geometry is simple: you project a vector x on the direction of a vector v, and the projected vector is the one from the origin to the base of the perpendicular from x on the direction v: vv^{*}x = vk, where k is the length of the projection (the coordinate of x in v direction), k=v^{*}x = |v|*|x|*cos(v,x)= x*cos (where |v|=1).

Now an oblique projection is defined as:

OP=WV^{*}, where V^{*}W=I. So the oblique projection of A is OP_{A}=V^{*}AW.

Can you please explain what an oblique projection is? Geometrically I mean. For a vector case first. I'm guessing the projection of vector x should be the vector in the direction of w but up to the point where the direction of v through x intersects the direction of w. But v^{*}w = I. I cannot see this in a plane .

Not to mention the projection of a matrix (the column vectors of it I guess). A drawing would be nice :D.

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# Oblique projection

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