Originally Posted by

**ygolo**
Well, in theory a 3-D ellipsoid has 8 degrees of freedom (3 degrees for the origin, two degrees of freedom for the three axis, and 3 for the length, width, height along those axis). 8 degrees of freedom,8 functions--seems plausible.

I was actually trying to think through the equations of projections of the ellipsoid on the various "plane vectors" representing function scores.

Any 3-D ellipsoid can be specified by a 3x3 positive definite matrix, B, (defines axis and "lengths" along those axes), and a 3-D vector, r, (defines the origin).

The bounds of the ellipsoid are defined by the solutions to the equation:

[(x-r)^T][B^-1](x-r)=1

Each cognitive function vector would have a 3x3 projection matrix, P, of rank 1. These matrices are symmetric and have the property that P^2=P.

Now the projection of the solutions to the equation above using P becomes the solutions to:

[(y-Pr)^T]P[B^-1](y-Pr)=1

I was thinking that I could make the function score of the ellipsoid to be given by the maximum valued solution to (8 versions of) the above equation.

For convenience, lets denote the Projection Matrix by the actual function name.

Si score = max y such that, [(y-Sir)^T]Si[B^-1](y-Sir)=1

Ni score = max y such that, [(y-Nir)^T]Ni[B^-1](y-Nir)=1

Se score = max y such that, [(y-Ser)^T]Si[B^-1](y-Ser)=1

Ne score = max y such that, [(y-Ner)^T]Ni[B^-1](y-Ner)=1

Fi score = max y such that, [(y-Fir)^T]Si[B^-1](y-Fir)=1

Ti score = max y such that, [(y-Tir)^T]Ni[B^-1](y-Tir)=1

Fe score = max y such that, [(y-Fer)^T]Si[B^-1](y-Fer)=1

Te score = max y such that, [(y-Ter)^T]Ni[B^-1](y-Ter)=1

Note: in all the above equations the "y" is independently bound, that is, each y is a different y (I just didn't want to do y_Si, y_Ni, etc.).

Other things to note:

-The directions of the axes of the ellipsoid are given by the eigenvectors of B, and the "half-axis length" along those axes is given by the square-root of the corresponding eigenvalues.

-B must be positive definite, and because of that, it must be symmetric (we're dealing completely with real-numbers here).

With the things noted above, we've now created a framework of 9 scalar variables.

--b1 b2 b3

B=b2 b4 b5

--b3 b5 b6

--r1

r=r2

--r3

Given the eight constraints above and one more for positive definiteness, it seemed like it was doable.

The issue is that I haven't yet thought thought what vectors should represent the functions, and I need to make sure that equations given from the projections are independent (or at least not contradictory).