I am working with polytopes that are defined by half-planes in [itex]\Re^N[/itex]. So they are defined by a number of inequalities (half plane representation), but can also be represented by the intersection points of these half planes (vertex representation). Computing the vertices is expensive, so I prefer to do that only when necessary.(adsbygoogle = window.adsbygoogle || []).push({});

I need to apply a transformation from this space into one of the same dimension. The transformation is just a linear one via matrix multiplication.

The matrix happens to be formed as a product of three non-square matrices. I could provide more detail about this, but in the end, the final matrix used for transportation (the product) is invertible.

The polytope I start with is convex. Is there a straightforward argument that shows this transformation will preserve that convexity? When I imagine a planar example, it seems like the stretching necessary to create concave dimples or change the angle of a corner point would have to be nonlinear. I also can't imagine a composition of shift, scale and rotation operations that would do this.

I would like to find an argument from first principles or a check to perform on the matrices. Any ideas or solutions would be appreciated.

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# Does multiplication by an invertible matrix preserve convexity?

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