### Linear Algebra for Graphics Geeks (X'PDP'X)

After realizing I'd already covered much of yesterday's post, I decided to send it to the bone pile.

And now today's experiment.

Finally, it's time to ask the question:

The answer is we have a matrix equation for an ellipse. It takes the following form:

X

Where Q corresponds to the (rotation * scale * rotation) in the final equation.

If theta is zero, the rotations become identity matrices, and it's easy to see that the product sans rotation is X

The next point to make is that Q is an eigen decomposition.

Q = PDP

And, there's a beautiful symmetry to it all.

X

The question I'll leave to the reader is:

This is post #13* of Linear Algebra for Graphics Geeks. Here are links to posts #1, #2, #3, #4, #5, #6, #7, #8, #9, #10, #11, #12.

Next, more commentary on the subject of eigenvectors and the SVD.

*I feel I should be putting a disclaimer on many of these posts. The impetus behind my recent posts on linear algebra is the desire for a cerebral form of urban renewal. It's time to raze the crumbling buildings with broken windows and rebuild from the ground up as time permits.*

And now today's experiment.

*Given an angle theta, the following transformation rotates the the vectors (cos theta, sin theta) and (-sin theta, cos theta) into alignment with the x and y axes.**Given constants A and B, scale the results of the previous transformation.**Next, reverse the first rotation, putting the x and y axes back into alignment with theta.**The net effect of these first three steps is scaling the world along the vector (cos theta, sin theta) and its normal vector (-sin theta, cos theta) by 1/A*

^{2}and 1/B^{2}, respectively.*Now multiply everything by (x,y) as a row vector.**Rearrange the parens, which can be done because matrix multiplication is associative.**Impose the constraint that entire product must equal 1.*Finally, it's time to ask the question:

**What have we here?**The answer is we have a matrix equation for an ellipse. It takes the following form:

X

^{T}QX = 1Where Q corresponds to the (rotation * scale * rotation) in the final equation.

If theta is zero, the rotations become identity matrices, and it's easy to see that the product sans rotation is X

^{2}/A^{2}+ Y^{2}/B^{2}= 1.The next point to make is that Q is an eigen decomposition.

Q = PDP

^{-1}And, there's a beautiful symmetry to it all.

X

^{T}(PDP^{-1})X = 1The question I'll leave to the reader is:

**What's the transform that maps the unit circle into the ellipse described by this equation?***Collecting that idea and all the other ideas into a single example was a major objective here.**P**lease excuse any redundancies in these posts. When I think up examples I find insightful and I feel the inclination, those examples very well may show up here.*This is post #13* of Linear Algebra for Graphics Geeks. Here are links to posts #1, #2, #3, #4, #5, #6, #7, #8, #9, #10, #11, #12.

Next, more commentary on the subject of eigenvectors and the SVD.

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