Consider the $2\times 2$ matrices \begin{align} A=\begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix} \end{align} and \begin{align} B=\begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{bmatrix} \end{align} and the vector \begin{align} u=\begin{bmatrix}\cos\theta \\ \sin\theta\end{bmatrix} \end{align} For all $\theta\in[0,2\pi]$, and given $A$ and $B$, consider the set of 2-D points whose co-ordinates are given as $(u^TAu,u^TBu)$, How do I plot this region?
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2 Answers
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Consider
A = {{1, 2}, {3, 4}};
B = {{5, 6}, {7, 8}};
Then
ParametricPlot[
{
{Cos[θ], Sin[θ]}.A.{Cos[θ], Sin[θ]},
{Cos[θ], Sin[θ]}.B.{Cos[θ], Sin[θ]}
},
{θ, 0, 2 π}]
Or more generally,
A = {{1, 2}, {3, 4}};
B = {{5, 6}, {x, 8}};
ParametricPlot[
Table[{{Cos[θ], Sin[θ]}.A.{Cos[θ],
Sin[θ]}, {Cos[θ], Sin[θ]}.B.{Cos[θ],
Sin[θ]}}, {x, -9, 9}] // Evaluate, {θ, 0,
2 π}, AspectRatio -> 1]
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$\begingroup$ I hope you don't mind the edit... $\endgroup$chris– chris2015-01-02 08:44:35 +00:00Commented Jan 2, 2015 at 8:44
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$\begingroup$ Probably, this is not the right place, but am just trying. Any explanation or thoughts as to why this looks like a ellipse? $\endgroup$dineshdileep– dineshdileep2015-01-02 16:41:47 +00:00Commented Jan 2, 2015 at 16:41
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1$\begingroup$ It is an ellipse. Here {x,y}.A.{x,y} is a quadratic form in x and y: a x^2 + b x + c y^2 + d6 + e, and likewise for matrix B. Thus the general solutions are the quadratics (circle, ellipse, hyperbola, etc.)... however for our case, {x,y} = {Cos[Theta],Sin[Theta]} and hence with bound norm of 1.0, restricting the solutions to generalized ellipses: ellipses, circles, (degenerate) lines, and (degenerate) points. Let A = B to see one degenerate case. $\endgroup$David G. Stork– David G. Stork2015-01-02 16:58:05 +00:00Commented Jan 2, 2015 at 16:58
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Any explanation or thoughts as to why this looks like a ellipse?
A = Array[a, {2, 2}];
B = Array[b, {2, 2}];
u = {Cos[t], Sin[t]};
({u.A.u, u.B.u} // TrigReduce // Simplify)
/. {a[i_, j_] :> Subscript[a, i, j], b[i_, j_] :> Subscript[b, i, j]}
$$\left\{\frac{1}{2} \left(\left(a_{1,2}+a_{2,1}\right) \sin (2 t)+\left(a_{1,1}-a_{2,2}\right) \cos (2 t)+a_{1,1}+a_{2,2}\right),\frac{1}{2} \left(\left(b_{1,2}+b_{2,1}\right) \sin (2 t)+\left(b_{1,1}-b_{2,2}\right) \cos (2 t)+b_{1,1}+b_{2,2}\right)\right\}$$
So the plot is a translated Lissajous figure with both frequencies equal to $2$. As the frequencies are equal, the figure is an ellipse.
ParametricPlot[{u.a.u, u.b.u}, {θ, 0, 2 π}]$\endgroup$