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I want to represent a rotated ellipse with matrices.

From these answers:

https://math.stackexchange.com/a/434482/197705

What is the parametric equation of a rotated Ellipse (given the angle of rotation)

I know one can describe a rotate ellipse. I know from this answer,

https://math.stackexchange.com/a/1403133/197705

it is possible but I don't understand how to do it.

How do you go from this

$$\dfrac {((x-h)\cos(A)+(y-k)\sin(A))^2}{(a^2)}+\dfrac{((x-h) \sin(A)-(y-k) \cos(A))^2}{(b^2)}=1,$$

to this

$$ (\mathbf{x}-\mathbf{v})^TA(\mathbf{x}-\mathbf{v})=1 $$

The notation is not consistent since these are quotes from two different answers.

I need to know this so I can understand some elliptical properties of the isocurve of bivariate normal distributions.

Stan Shunpike
  • 4,921

1 Answers1

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Let $\mathbf{x} = \begin{bmatrix}x_1 \\ x_2\end{bmatrix}$ and $\mathbf{v} = \begin{bmatrix}v_1 \\ v_2\end{bmatrix}$.

$$\left(\frac{(x_1-v_1) \cos \theta + (x_2 - v_2) \sin \theta}{a}\right)^2 + \left(\frac{(x_1-v_1) \sin \theta - (x_2 - v_2) \cos \theta}{b}\right)^2 = (\mathbf{x-v})^\top \begin{bmatrix}\cos \theta & \sin \theta \\ \sin \theta & - \cos \theta \end{bmatrix}^\top \begin{bmatrix} a^{-2} \\ & b^{-2} \end{bmatrix} \begin{bmatrix}\cos \theta & \sin \theta \\ \sin \theta & - \cos \theta \end{bmatrix} (\mathbf{x-v}).$$

angryavian
  • 89,882
  • What's the name for the type of decomposition you did with the rotational matrices? I want to understand the direction of the major and minor axes and assume eigenvectors are needed for that? – Stan Shunpike Jan 18 '20 at 23:37