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- Summary:
- I modeled an ellipse of the form x^2 + 2bx + 2cxy + 2dy + ey^2=1 with a 3x3 matrix and am unsure what the meaning of the third eigenvalue is, since there are only two axes.

While reading the Strang textbook on tilted ellipses in the form of ax^2+2bxy +cy^2=1, I got to thinking about ellipses of the form ax^2 + 2bx + 2cxy + 2dy + ey^2=1 and wondered if I could model them through 3x3 symmetric matrices. I think I figured out something that worked for

Anyway, my question involves the meaning of the three eigenvalues. In the 2x2 case, they are the semi-major and semi-minor axes. What about in this 3x3 case? I would assume two of them are again related to the axes, but what about the third? Or perhaps I just made up a procedure that was non-sensical.

Also, I assume this is a linear algebra topic, but if it would be better somewhere else, please let me know.

**xT**A**x**, where**x**= (x, y, 1). For example, x^2+4x+6xy+8y+2y^2=1 has matrix A with row 1= 1,3,2; row 2= 3,2,4; row 3=2,4,0Anyway, my question involves the meaning of the three eigenvalues. In the 2x2 case, they are the semi-major and semi-minor axes. What about in this 3x3 case? I would assume two of them are again related to the axes, but what about the third? Or perhaps I just made up a procedure that was non-sensical.

Also, I assume this is a linear algebra topic, but if it would be better somewhere else, please let me know.