Tag Archives: quadratic form

Fun with shear operations and SVD – V – matrices of sheared n-dimensional ellipsoids

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In my previous post of the series

Fun with shear operations and SVD – I – shear matrices and examples created with Blender
Fun with shear operations and SVD – II – Shearing of rectangles and cubes with Python and Matplotlib
Fun with shear operations and SVD – III – Shearing of circles
Fun with shear operations and SVD – IV – Shearing of ellipses

we have studied the transformation of an ellipse by a shear operation. The coordinates of points on an ellipse and the components of respective position vectors fulfill a quadratic equation (quadratic form):

\[
\alpha_o\,x_o^2 \, + \, \beta_o \, x_o y_o \, + \, \gamma_o \, y_o^2 \:=\: \delta_o
\]

An equivalent matrix equation for respective vectors \( \left(\,x_o,\, y_o\,\right)^T \) is

\[
\left(\,x_o,\, y_o\,\right) \,\circ\, \pmb{\operatorname{A}}_q^O \,\circ\, \left(\,x_o,\, y_o\,\right)^T \: = \: \delta_o
\]

The superscript “T” symbolizes the transposition operation. The symmetric (2×2)-matrix \( \pmb{\operatorname{A}}_q^O \) defines the original, unsheared ellipse. The suffix “q” indicates the quadratic form. I have shown how the shear parameter λS impacts the coefficients of a corresponding (2×2)-matrix \(\pmb{\operatorname{A}}_q^S \) that defines the sheared ellipse.

What I have not done in the last post is to show how our matrix \(\pmb{\operatorname{A}}_q^S \) is related to a shear matrix \(\pmb{\operatorname{M}}_{sh} \) (see the first post), which describes the effect of the shear on the vectors \( \left(\,x_o,\, y_o\,\right)^T \). I am going to discuss this below. The given matrix relations will also be valid for general n-dimensional ellipsoids.

Matrix relations as discussed below are helpful to accelerate numerical calculations as Numpy (in cooperation with libraries for your OS) provides highly optimized modules for matrix operations. n-dimensional ellipsoids furthermore characterize hyper-surfaces of multivariate normal distributions which appear in certain areas of Machine Learning and respective data.

Matrix describing a centered n-dimensional ellipsoid

We consider n-dimensional and centered ellipsoids whose symmetry centers coincide with the origin of the Euclidean coordinate system [ECS] we work with. A position vector \(\left(\,x_1^o,\, x_2^o\, \cdots x_n^o \right)^T \)

\[
\pmb{x_o} \: = \: \begin{pmatrix} x_1^o \\ x_2^o \\ \vdots \\ x_n^o \end{pmatrix}
\]

is a vector drawn from the origin to a point on the ellipsoid’s hyper-surface. Note that a general vector of a vector space has no reference to a coordinate system’s origin. Therefore the distinction. A general ellipsoid is defined by a quadratic form in the components of its position vectors. The quadratic form is equivalent to the following matrix equation

\[
\left(\pmb{x_o}\right)^T \,\circ\, \pmb{\operatorname{A}}_{qn}^O \,\circ\, \pmb{x_o} \: = \: 1
\]

where \(\pmb{\operatorname{A}}_{qn}^O \) now represents a symmetric (nxn)-matrix. The “\( \circ \)” symbolizes a matrix product.

Note: A coefficient \( \delta \gt 0 \) which we have used in previous posts on the right side of the equation can be included in the coefficient values of the matrix).

Note that the equations above define an ellipsoid up to a translation vector. This is reflected in the fact that the above equation does not create any linear terms.

Quadratic forms not only define ellipsoids. For an ellipsoid we have to assume that the determinant of\(\pmb{\operatorname{A}}_{qn}^O \) is > 0 and that the matrix is invertible:

\[
\operatorname{det} \left(\pmb{\operatorname{A}}_{qn}^O \right) \: \gt \: 0
\]

Note that you could choose an ECS in which the ellipsoid’s principal axes would align with the ECS’s coordinate axes. Such a choice would correspond to a PCA-transformation of the vector data. \(\pmb{\operatorname{A}}_{qn}^O \) would then become diagonal. This corresponds to the fact that a symmetric matrix always has an eigenvalue-decomposition.

Equation of the quadratic form for the sheared ellipsoid

In the first post of this series I have defined a (invertible) shear matrix as a unipotent matrix \( \pmb{\operatorname{M}}_{sh} \) with all coefficients of the lower triangular part, off the diagonal, being equal to 0.0 and all elements on the diagonal being equal to 1:

\[ \pmb{\operatorname{M}}_{sh} \, = \,
\begin{pmatrix}
1 & m_{12} &\cdots & m_{1n}(\ne0) \\
0 & 1 &\cdots & m_{2n} \\
\vdots &\vdots &\ddots &\vdots \\
0 & 0 &\cdots & 1 \end{pmatrix}
\]

Note:

\[ \operatorname{det} \left(\pmb{\operatorname{M}}_{sh}\right) : = \: 1
\]

So an inverse matrix \( \pmb{\operatorname{M}}_{sh}^{-1} \) exists. Shearing our original ellipse (with position vectors \( \pmb{x_o} \)) leads to new vectors \( \pmb{x_S} \):

\[
\pmb{x_S} \: =\: \,\pmb{\operatorname{M}}_{sh} \,\circ\, \pmb{x_o}
\]

We insert \( \pmb{x_S} \) into our defining equation of the original ellipsoid to derive a matrix equation for the sheared ellipsoid:

\[
\left[ \, \pmb{\operatorname{M}}_{sh}^{-1} \,\circ\, \pmb{x_S} \, \right]^T \,\circ\, \pmb{\operatorname{A}}_{qn}^O \,\circ\, \left[ \, \pmb{\operatorname{M}}_{sh}^{-1} \,\circ\, \pmb{x_S} \, \right] \: = \: 1
\]

Giving:

\[
\left(\pmb{x_S}\right)^T \,\circ\, \left[ \, \left( \, \pmb{\operatorname{M}}_{sh}^{-1} \, \right)^T \,\circ\, \pmb{\operatorname{A}}_{qn}^O \,\circ\, \, \pmb{\operatorname{M}}_{sh}^{-1} \, \right] \,\circ\, \pmb{x_S} \: = \: 1
\]

This, obviously, is a new definition equation for a quadratic form in the components of \( \pmb{x_S} \) with a matrix

\[
\pmb{\operatorname{A}}_{qn}^S \:=\: \left( \, \pmb{\operatorname{M}}_{sh}^{-1} \, \right)^T \,\circ\, \pmb{\operatorname{A}}_{qn}^O \,\circ\, \, \pmb{\operatorname{M}}_{sh}^{-1}
\]

We also find:

\[
\operatorname{det}\left(\pmb{\operatorname{A}}_{qn}^S \right) \:=\: \operatorname{det}\left(\pmb{\operatorname{A}}_{qn}^O \right) \: \gt \: 0
\]

From this we can conclude with confidence that we again have gotten a n-dimensional ellipsoid.

Inclusion of a SVD eigendecomposition of MS

A “Singular Value Decomposition” [SVD] can be applied to any (nxm)-matrix Q (with n > m):

\[
\pmb{\operatorname{Q}} \:=\: \pmb{\operatorname{U}} \,\circ\, \pmb{\operatorname{\Sigma}} \,\circ\, \, \pmb{\operatorname{V}}^T
\]

The (nxn)-matrix U and the (mxm)-matrix V are orthonormal matrices:

\[ \begin{align}
\pmb{\operatorname{U}} \,\circ\, \pmb{\operatorname{U}}^T \:&=\: 1 \\
\pmb{\operatorname{V}} \,\circ\, \pmb{\operatorname{V}}^T \:&=\: 1
\end{align}
\]

Σ is a diagonal (nxm)-matrix with singular values. The column-vectors of U and V are orthogonal singular vectors. Geometrically, U and V can be interpreted s rotational operations.

Therefore, we can decompose a (nxn) upper triangular shear-matrix into two orthonormal (nxn)-matrices U and V plus a diagonal matrix Σ :

\[
\pmb{\operatorname{M}}_{sh} \:=\: \pmb{\operatorname{U}} \,\circ\, \pmb{\operatorname{\Sigma}} \,\circ\, \, \pmb{\operatorname{V}}^T
\]

This leads to

\[ \begin{align}
\pmb{\operatorname{M}}_{sh}^{-1} \:&=\: \left[\pmb{\operatorname{V}}^T\right]^{-1} \,\circ\, \pmb{\operatorname{\Sigma}}^{-1} \,\circ\, \, \pmb{\operatorname{U}}^{-1} \\
&=\: \pmb{\operatorname{V}} \,\circ\, \pmb{\operatorname{\Sigma}}^{-1} \,\circ\, \pmb{\operatorname{U}}^T
\end{align}
\]

This gives us an alternative form to define the inverse shear matrix. Note that the order of the matrices in the matrix products is essential.

An example for the case of a sheared ellipse

We use the example of a sheared ellipse discussed in the last post to verify the results above numerically for a 2-dim case. To write a respective Python/Numpy-program is simple. I will just give you my numerical results below.

We have used an ellipse with the longer and shorter primary axes having values a = 2 and b = 1, respectively. The ellipse was rotated by 60° against the ECS-axes.

The respective (2×2)-matrix \( \pmb{\operatorname{A}}_q^O \) had the following coefficients

\[
\pmb{\operatorname{A}}_q^O \: = \: \begin{pmatrix} \alpha_o & 1/2\, \beta_o \\ 1/2\,\beta_o & \gamma_o \end{pmatrix} \:=\:
\begin{pmatrix} 3.25 & -1.29903811 \\ -1.29903811 & 1.75 \end{pmatrix}
\]

to fulfill

\[
\alpha_o\,x_o^2 \, + \, \beta_o \, x_o y_o \, + \, \gamma_o \, y_o^2 \:=\: \delta_o \:=\: 4.0
\]

The shear matrix (with λS = 0.6) was

\[
\pmb{\operatorname{M}}_{sh} \:=\: \begin{pmatrix} 1.0 & 0.6 \\ 0.0 & 1.0 \end{pmatrix}
\]

The resulting sheared ellipse became

For an ellipse we have shown that \(\pmb{\operatorname{A}}_q^S \) is given by

\[
\pmb{\operatorname{A}}_q^S \: = \: \begin{pmatrix} \alpha_o & 1/2\,\left(\beta_o \,-\,2 \alpha_o \lambda_S \right) \\
1/2\,\left(\beta_o \,-\,2 a_o \lambda_S \right) & \alpha_o \, \lambda_S^2 \, -\, \beta_o \lambda_S \,+\, \gamma_o \end{pmatrix}
\]

From this we get the following numerical values:

A_q^S = 
 [[ 3.25       -3.24903811]
 [-3.24903811  4.47884573]]

Via the Python-statement 

M_sh_inv = np.linalg.inv(M_sh)

and 

A_q^S_2 = M_sh_inv.T @ A_q @ M_sh_inv

we get the following values 

A_q^S_2 = 
 [[ 3.25       -3.24903811]
 [-3.24903811  4.47884573]]

Identical! Using 

U_sh, S, Vt_sh = np.linalg.svd(M_sh, full_matrices=True)
S_sh = np.diag(S)
M_sh_2 = U_sh @ S_sh @ Vt_sh
M_sh_inv_2 = Vt_sh.T @ np.linalg.inv(S_sh) @ U_sh.T
A_q^S_2 = M_sh_inv_2.T @ A_q @ M_sh_inv_2

we also reproduce the exactly same values.

Conclusion

We have shown how a shear matrix \( \pmb{\operatorname{M}}_{sh} \) transforms the matrix \(\pmb{\operatorname{A}}_{qn}^O \) which defines an un-sheared n-dimensional ellipsoid into a matrix \(\pmb{\operatorname{A}}_{qn}^S \) defining its sheared counterpart. We have also had a glimpse on a SVD decomposition of a shear matrix. The results will enable us in the next post to apply shear operations on a concrete example of a 3-dimensional ellipsoid.

 

Properties of ellipses by matrix coefficients – II – coordinates of points with extremal y-values

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In the previous post of this mini-series

Properties of ellipses by matrix coefficients – I – Two defining matrices

I have discussed how the coefficients of two matrices which can be used to define a centered, rotated ellipse can be used to calculate geometrical properties of the ellipse:

The lengths σ1, σ2 of the ellipse’s principal axes and the rotation angle by which the major axis is rotated against the x-axis of the Cartesian coordinate system [CCS] we work with.

But there are other properties which are interesting, too. A centered, rotated ellipse has two points with extremal values in their y-coordinates. Can we express the coordinates – or equivalently the components of respective position vectors – in terms of the basic matrix coefficients?

The answer is, of course, yes. This post provides a derivation of respective formulas.

Matrix equation for an ellipse

In the last post we have shown that a centered ellipse is defined by a quadratic form, i.e. by a polynomial equation with quadratic terms in the components xE and yE of position vectors for points of the ellipse:

\[
\alpha * x_E^2 \, + \, \beta * x_E * y_E \, + \, \gamma * y_E^2 \:=\: 1 \,.
\]

The quadratic polynomial can be formulated as a matrix operation applied to position vectors vE of points on an ellipse. With the the quadratic and symmetric matrix Aq

\[ \pmb{\operatorname{A}}_q \:=\:
\begin{pmatrix} \alpha & \beta / 2 \\ \beta / 2 & \gamma \end{pmatrix}
\]

we can rewrite the polynomial equation for the centered ellipse as

\[
\pmb{v}_e^T \circ \pmb{\operatorname{A}}_q \circ \pmb{v}_E \:=\: 1 \,, \quad \operatorname{with}\: \pmb{v_E} \,=\, \begin{pmatrix} x_E \\ y_E \end{pmatrix}.
\]

We know already that (under standard conditions)

\[
\operatorname{det}\left( \pmb{\operatorname{A}}_q \right) \:=\: \left( \, \alpha \, \gamma \,-\, {1\over 4}\, \beta^2\, \right) \: \gt \: 0 \,.
\]

Aq is a symmetric and invertible matrix.

Points of the ellipse with extremal yE – values

Our first step to get the xE and yE coordinates for extremal points is to evaluate the quadratic form with respect to yE. We define:

\[ \begin{align}
a_h \,& =\, {\alpha \over \gamma} \,, \\[8pt]
b_h \,& =\, {1 \over 2 } {\beta \over \gamma} \,, \\[8pt]
d_h \,& =\, {1 \over \gamma} \,.
\end{align}
\]

We reorder terms of the equation and add a supplemental quadratic term on both sides:

\[
y_E^2 \, +\, b_h\,x_E\,y_E \, +\, b_h^2 \, x_E^2 \:=\: d_h \,-\, a_h \, x_E^2 \,+\, b_h^2 \, x_E^2 \,.
\]

We evaluate the complete quadratic term ob the left side to get

\[
y_E \:=\: – b_h \, x_E \, \pm \, \left[\,d_h \,-\, \left( a_h \,-\, b_h^2 \right)\, x_E^2 \, \right]^{1/2} \,.
\]

Let us first focus on the positive of the two alternative terms:

\[
y_E \:=\: – b_h \, x_E \,+\, \left[\,d_h \,-\, \left( a_h \,-\, b_h^2 \right)\, x_E^2 \, \right]^{1/2} \,.
\]

We get an extremal yE by evaluating the derivative with respect to xE

\[
{ \partial \, y_E \over \partial \, x_E } \:=\: 0 \,.
\]

This means

\[
– b_h \, – \, { \left(a_h \,-\, b_h^2 \right)\,x_E \over \left[\,d_h \,-\, \left( a_h \,-\, b_h^2 \right)\, x_E^2 \, \right]^{1/2} } \:=\: 0 \,.
\]

Getting rid of the denominator gives

\[
– b_h \, \left[\,d_h \,-\, \left( a_h \,-\, b_h^2 \right)\, x_E^2 \, \right]^{1/2} \, – \, \left(a_h \,-\, b_h^2 \right)\,x_E \:=\: 0
\]

and

\[
x_E \:=\: { – b_h \, \left[\,d_h \,-\, \left( a_h \,-\, b_h^2 \right)\, x_E^2 \, \right]^{1/2} \over \left(a_h \,-\, b_h^2 \right) } \,.
\]

Squaring both sides and reordering leads to

\[
x_E^2 \:=\: { – b_h^2 \, d_h \over a_h \, \left(a_h \,-\, b_h^2 \right) } \,.
\]

Using the old coefficients provides the final result for xe:

\[
x_E \:=\: \pm \, {1 \over 2} \, \beta \, \left[ { 1 \over \alpha \,\left( \alpha \, \gamma \,-\, {1 \over 4} \beta^2 \right) } \right]^{1/2} \,.
\]

We now follow the alternative solution for yE (see above). After a calculation of the yE-values from the derived xE values, we get the components for the two position vectors to the points with extremal y-values on the ellipse:

\[ \begin{align}
x_{E1}^{max} \:&=\: – \, {1 \over 2} \, \beta \, \left[ { 1 \over \alpha \,\left( \alpha \, \gamma \,-\, {1 \over 4} \beta^2 \right) } \right]^{1/2} \,,
\\[8pt]
y_{E1}^{max} \:&=\: – \, {1 \over 2} \, {\beta \over \gamma} \, x_{E1}^{max} \,+\,
\left[ { 1 \over \gamma} \,-\, \left( {\alpha \over \gamma} \,-\, {1 \over 4} {\beta^2 \over \gamma^2} \right) \, x_{E1}^{max} \right]^{1/2}
\end{align}
\]

and

\[ \begin{align}
x_{E2}^{max} \:&=\: + \, {1 \over 2} \, \beta \, \left[ { 1 \over \alpha \,\left( \alpha \, \gamma \,-\, {1 \over 4} \beta^2 \right) } \right]^{1/2} \,, \\[8pt]
y_{E2}^{max} \:&=\: + \, {1 \over 2} \, {\beta \over \gamma} \, x_{E2}^{max} \,-\,
\left[ {1 \over \gamma} \,-\, \left( {\alpha \over \gamma} \,-\, {1 \over 4} {\beta^2 \over \gamma^2} \right) \, x_{E2}^{max} \right]^{1/2} \,.
\end{align}
\]

Solution in terms of the coefficients of an alternative matrix AE

In my previous post I have discussed yet another matrix AE which also can be used to define an ellipse. This matrix summarizes two affine transformations of a centered unit circle: AE = Rφ ○ Dσ1, σ2. Dσ1, σ2.

You find the relations between the coefficients (a, b, c, d) of matrix AE and the coefficients (α, β, γ) of matrix Aq in my previous post. This will allow you to calculate the vectors to the extremal points of an ellipse in terms of the coefficients (a, b, c, d).

Conclusion

In this post we have again used the coefficients of a matrix which defines an ellipse via a quadratic form to get information about a geometrical property.
We can now calculate the components of the position vectors to the two points of an ellipse with extremal y-values as functions of the matrix coefficients.

In the next post

Properties of ellipses by matrix coefficients – III – coordinates of points with extremal radii

I will show you how to calculate the components of the end points of the principal axes of the ellipse with the help of our matrix for a quadratic form. I will also use our theoretical results for plots of some ellipses’ axes and of their extremal points. We will also compare theoretical predictions with numerically evaluated values.

 

Properties of ellipses by matrix coefficients – I – Two defining matrices and eigenvalues

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For my two current post series on multivariate normal distributions [MNDs] and on shear operations

Multivariate Normal Distributions – II – random vectors and their covariance matrix

Fun with shear operations and SVD

some geometrical and algebraic properties of ellipses are of interest.

Geometrically, we think of an ellipse in terms of their perpendicular two principal axes, focal points, ellipticity and rotation angles with respect to a coordinate system. As elliptic data appear in many contexts of physics, chaos theory, engineering, optics … ellipses are well studied mathematical objects. So, why a post about ellipses in the Machine Learning section of a blog?

In my present working context ellipses appear as a side result of statistical multivariate normal distributions [MNDs]. The projections of multidimensional contour hyper-surfaces of a MND within the ℝn onto coordinate planes of an Cartesian Coordinate System [CCS] result in 2-dimensional ellipses. These ellipses are typically rotated against the axes of the CCS – and their rotation angles reflect data correlations. The general relations of statistical vector data with projections of multidimensional MNDs are somewhat intricate.

Data produced in numerical experiments, e.g. in a Machine Learning context, most often do not give you the geometrical properties of ellipses, which some theory may have predicted, directly. Instead you may get numerical values of statistical vector distributions which correspond to algebraic coefficients. Examples of such coefficients are e.g. correlation coefficients. These coefficients can often be regarded as elements of a matrix. In case of an underlying MND of your statistical variables these matrices indirectly and approximately describe contour surfaces – namely ellipsoids. Or regarding projections of the data on 2-dim coordinate planes “ellipses”.

Ellipsoids/ellipses can in general be defined by matrices operating on position vectors. In particular: Coefficients of quadratic polynomial expressions used to describe ellipses as conic sections correspond to the coefficients of a matrix operating on position vectors.

So, when I became confronted with multidimensional MNDs and their projections on coordinate planes the following questions became interesting:

  • How can one derive the lengths σ1, σ2 of the perpendicular principal axes of an ellipse from data for the coefficients of a matrix which defines the ellipse by a polynomial expression?
  • By which formula do the matrix coefficients provide the inclination angle of the ellipse’s primary axes with the x-axis of a chosen coordinate system?

You may have to dig a bit to find correct and reproducible answers in your math books. Regarding the resulting mathematical expressions I have had some bad experiences with ChatGPT. But as a former physicist I take the above questions as a welcome exercise in solving quadratic equations and doing some linear algebra. So, for those of my readers who are a bit interested in elementary math I want to answer the posed questions step by step and indicate how one can derive the respective formulas. The level is moderate – you need some knowledge in trigonometry and/or linear algebra.

Centered ellipses and two related matrices

Below I regard ellipses whose centers coincide with the origin of a chosen CCS. For our present purpose we thus get rid of some boring linear terms in the equations we have to solve. We do not loose much of general validity by this step: Results of an off-center ellipse follow from applying a simple translation operation to the resulting vector data. But I admit: Your (statistical) data must give you some relatively precis information about the center of your ellipse. We assume that this is the case.

Our ellipses can be rotated with respect to a chosen CCS. I.e., their longer principal axes may be inclined by some angle φ towards the x-axis of our CCS.

There are actually two different ways to define a centered ellipse by a matrix:

  • Alternative 1: We define the (rotated) ellipse by a matrix AE which results from the (matrix) product of two simpler matrices: AE = Rφ ○ Dσ1, σ2.
    Dσ1, σ2 corresponds to a scaling operation applied to position vectors for points located on a centered unit circle. Aφ describes a subsequent rotation. AE summarizes these geometrical operations in a compact form.
  • Alternative 2: We define the (rotated) ellipse by a matrix Aq which combines the x- and y-elements of position vectors in a polynomial equation with quadratic terms in the components (see below). The matrix defines a so called quadratic form. Geometrically interpreted, a quadratic form describes an ellipse as a special case of a conic section. The coefficients of the polynomial and the matrix must, of course, fulfill some particular properties.

While it is relatively simple to derive the matrix elements from known values for σ1, σ2 and φ it is a bit harder to derive the ellipse’s properties from the elements of either of the two defining matrices. I will cover both matrices in this post.

For many practical purposes the derivation of central elliptic properties from given elements of Aq is more relevant and thus of special interest in the following discussion.

Matrix AE of a centered and rotated ellipse: Scaling of a unit circle followed by a rotation

Our starting point is a unit circle C whose center coincides with our CCS’s origin. The components of vectors vc to points on the circle C fulfill the following conditions:

\[
\pmb{C} \::\: \left\{ \pmb{v}_c \:=\: \begin{pmatrix} x_c \\ y_c \end{pmatrix} \:=\: \begin{pmatrix} \operatorname{cos}(\psi) \\ \operatorname{sin}(\psi) \end{pmatrix}, \quad 0\,\leq\,\psi\, \le 2\pi \right\}
\]

and

\[
x_c^2 \, +\, y_c^2 \,=\; 1 \,.
\]

We define an ellipse E(σ1, σ2) by the application of two linear operations to the vectors of the unit circle:

\[
\pmb{E}_{\sigma_1, \, \sigma_2} \::\: \left\{ \, \pmb{v}_E \:=\: \begin{pmatrix} x_E \\ y_E \end{pmatrix} \:=\: \pmb{\operatorname{R}}_{\phi} \circ \pmb{\operatorname{D}}_E \circ \pmb{v}_c , \quad \pmb{v_c} \in \pmb{C} \, \right\} \,.
\]

DE is a diagonal matrix which describes a stretching of the circle along the CCS-axes, and Rφ is an orthogonal rotation matrix. The stretching (or scaling) of the vector-components is done by

\[
\pmb{\operatorname{D}}_E \:=\: \begin{pmatrix} \sigma_1 & 0 \\ 0 & \sigma_2 \end{pmatrix} \,,
\]
\[
\pmb{\operatorname{D}}_E^{-1} \:=\: \begin{pmatrix} {1 / \sigma_1} & 0 \\ 0 & {1 / \sigma_2} \end{pmatrix} \,.
\]

The coefficients σ1, σ2 obviously define the lengths of the principal axes of the yet unrotated ellipse. To be more precise: σ1 is half of the diameter in x-direction, σ1 is half of the diameter in y-direction. I.e. σ1 and σ1 refer to the half-axes of the ellipse.

The subsequent rotation by an angle φ against the x-axis of the CCS is done by

\[
\pmb{\operatorname{R}}_{\phi} \:=\:
\begin{pmatrix} \operatorname{cos}(\phi) & – \,\operatorname{sin}(\phi) \\ \operatorname{sin}(\phi) & \operatorname{cos}(\phi)\end{pmatrix}
\:=\: \begin{pmatrix} u_1 & -\,u_2 \\ u_2 & u_1 \end{pmatrix} \,,
\]
\[
\pmb{\operatorname{R}}_{\phi}^T \:=\: \pmb{\operatorname{R}}_{\phi}^{-1} \:=\: \pmb{\operatorname{R}}_{-\,\phi} \,\,.
\]

The combined linear transformation results in a matrix AE with coefficients ((a, b), (c, d)):

\[ \begin{align}
\pmb{\operatorname{A}}_E \:=\: \pmb{\operatorname{R}}_{\phi} \circ \pmb{\operatorname{D}}_E \:=\:
\begin{pmatrix} \sigma_1\,u_1 & -\,\sigma_2\,u_2 \\ \sigma_1\,u_2 & \sigma_2\,u_1 \end{pmatrix} \:=\:: \begin{pmatrix} a & b \\ c & d \end{pmatrix} \,\,.
\end{align}
\]

These is the first set of matrix coefficients we are interested in.

Note:

\[ \pmb{\operatorname{A}}_E^{-1} \:=\:
\begin{pmatrix} {1 \over \sigma_1} \,u_1 & {1 \over \sigma_1}\,u_2 \\ -{1 \over \sigma_2}\,u_2 & {1 \over \sigma_2}\,u_1 \end{pmatrix} \, ,
\]
\[
\pmb{v}_E \:=\: \begin{pmatrix} x_E \\ y_E \end{pmatrix} \:=\: \pmb{\operatorname{A}}_E \circ \begin{pmatrix} x_c \\ y_c \end{pmatrix} \,,
\]
\[
\pmb{v}_c \:=\: \begin{pmatrix} x_c \\ y_c \end{pmatrix} \:=\: \pmb{\operatorname{A}}_E^{-1} \circ \begin{pmatrix} x_E \\ y_E \end{pmatrix} \, .
\]

We use

\[ \begin{align}
&u_1 \,=\, \operatorname{cos}(\phi),\quad u_2 \,=\,\operatorname{sin}(\phi), \quad u_1^2 \,+\, u_2^2 \,=\, 1 \,,\\[8pt]
&(1/\lambda_1) \,: =\, \sigma_1^2, \quad\quad (1/\lambda_2) \,: =\, \sigma_2^2
\end{align}
\]

and find

\[ \begin{align}
a \,&=\, \sigma_1\,u_1, \quad b \,=\, -\, \sigma_2\,u_2 \,, \\[8pt]
c \,&=\, \sigma_1\,u_2, \quad d \,=\, \sigma_2\,u_1 \,,
\end{align}
\]
\[
\operatorname{det}\left( \pmb{\operatorname{A}}_E \right) \:=\: a\,d \,-\, b\,c \:=\: \sigma_1\, \sigma_2 \,.
\]

σ1 and σ2 are factors which give us the lengths of the principal axes of the ellipse. σ1 and σ2 have positive values. We, therefore, can safely assume:

\[
\operatorname{det}\left( \pmb{\operatorname{A}}_E \right) \:=\: a\,d \,-\, b\,c \:\gt\: 0
\]

Ok, we have defined an ellipse via an invertible matrix AE, whose coefficients are directly based on geometrical properties.

But as said: Often an ellipse is described by a an equation with quadratic terms in x and y coordinates of data points. The quadratic form has its background in algebraic properties of conic sections. As a next step we derive such a quadratic equation and relate the coefficients of the quadratic polynomial with the elements of our matrix AE. The result will in turn define another very useful matrix Aq.

Quadratic forms – Case 1: Centered ellipse, principal axes aligned with CCS-axes

We start with a simple case. We take a so called axis-parallel ellipse which results from a scaling matrix DE, only. I.e., in this case, the rotation matrix is assumed to be just the identity matrix. We can omit it from further calculations:

\[
\pmb{\operatorname{R}}_{\phi} \:=\:
\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \,=\, \pmb{\operatorname{I}}, \quad u_1 \,=\, 1,\: u_2 \,=\, 0, \: \phi = 0 \,.
\]

We need an expression in terms of (xE, yE).

To get quadratic terms of vector components it often helps to invoke a scalar product. The scalar product of a vector with itself gives us the squared norm or length of a vector. In our case the norms of the inversely re-scaled vectors obviously have to fulfill:

\[
\left[\, \pmb{\operatorname{D}}_E^{-1} \circ \begin{pmatrix} x_E \\ y_E \end{pmatrix} \, \right]^T \,\bullet \, \left[\, \pmb{\operatorname{D}}_E^{-1} \circ \begin{pmatrix} x_E \\ y_E \end{pmatrix} \,\right] \:=\: 1 \,\,.
\]

(The bullet represents the scalar product of the vectors.) This directly results in:

\[
{1 \over \sigma_1^2} * x_E^2 \, + \, {1 \over \sigma_2^2} * y_E^2 \:=\: 1 \,.
\]

We eliminate the denominator to get a convenient quadratic form:

\[
\lambda_1 * x_E^2 \,+\, \lambda_2 * y_E^2 \:=1 \,.
\]

If we were given the quadratic form more generally by coefficients α, β and γ

\[
\alpha * x_E^2 \,+\, \beta * x_E \, y_E \,+\, \gamma * y_E^2 \,:=\, 1 \,,
\]

we could directly relate these coefficients with the geometrical properties of our ellipse:

Axis-parallel ellipse:

\[ \begin{align}
\alpha \,&=\, 1 / a^2 \,=\, 1 / \sigma_1^2 \,=\, \lambda_1 \,, \\[8pt]
\gamma \,&=\, 1 / d^2 \,=\, 1 / \sigma_2^2 \,=\, \lambda_2 \,, \\[8pt]
\beta \,&=\, b \,=\, c \,=\, 0 \,, \\[8pt]
\phi &= 0 \,.
\end{align}
\]

I.e., we can directly derive σ1, σ2 and φ from the coefficients of the quadratic form. But an axis-parallel ellipse is a very simple ellipse. Things get more difficult for a rotated ellipse.

Quadratic forms – Case 2: General centered and rotated ellipse

We perform the same trick with the vectors vE to get a quadratic polynomial for a rotated ellipse:

\[
\left[ \,\pmb{\operatorname{A}}_E^{-1} \circ \begin{pmatrix} x_E \\ y_E \end{pmatrix} \, \right]^T \, \bullet \,
\left[ \, \pmb{\operatorname{A}}_E^{-1} \circ \begin{pmatrix} x_E \\ y_E \end{pmatrix} \,\right] \:=\: 1 \,\,.
\]

I skip the lengthy, but simple algebraic calculation. We get (with our matrix elements a, b, c, d):

\[
{1 \over \sigma_1^2 \, \sigma_2^2 } \, \left[ \,
\left( c^2 \,+\, d^2 \right) * x_E^2 \,\, – \,\, 2\left( a\,c\, +\, b\,d \right)* x_E \, y_E \,\, + \,\, \left(a^2 \,+\, b^2\right) * y_E^2 \, \right]
\,=\, 1 \, .
\]

The rotation has obviously lead to mixing of components in the polynomial. The coefficient for xE * yE is > 0 for the non-trivial case.

Quadratic form: A matrix equation to define an ellipse

We rewrite our equation again with general coefficients α, β and γ

\[
\alpha * x_E^2 \, + \, \beta * x_E \, y_E \, + \, \gamma * y_E^2 \,=\, 1 \,.
\]

These are coefficients which may come from some theory or from averages of numerical data.

The quadratic polynomial can in turn be reformulated as a matrix operation with a symmetric matrix Aq:

\[
\pmb{v}_E^T \circ \pmb{\operatorname{A}}_q \circ \pmb{v}_E \:=\: 1 \,,
\]

with

\[ \pmb{\operatorname{A}}_q \:=\:
\begin{pmatrix} \alpha & \beta / 2 \\ \beta / 2 & \gamma \end{pmatrix} \,\,,
\]
\[ \pmb{\operatorname{A}}_q \:=\: { 1 \over \sigma_1^2 \, \sigma_2^2} \,
\begin{pmatrix} c^2 \,+\, d^2 & a\,c \, +\, b\,d \\ a\,c \, +\, b\,d & a^2 \,+\, b^2 \end{pmatrix} \, .
\]

This means

\[ \begin{align}
\alpha \:&=\: { 1 \over \sigma_1^2 \, \sigma_2^2} \, \left(\, c^2 \,+\, d^2 \,\right) \:=\: \lambda_2 * u_2^2 \, + \, \lambda_1 * u_1^2 \,, \\[8pt]
\gamma \:&=\: { 1 \over \sigma_1^2 \, \sigma_2^2} \, \left(\, a^2 \,+\, b^2 \,\right) \:=\: \lambda_2 * u_1^2 \, + \, \lambda_1 * u_2^2 \,, \\[8pt]
\beta \:&=\: – \, 2\, { 1 \over \sigma_1^2 \, \sigma_2^2} \, \left(\,a \,c \,+\, b \, d \,\right) \:=\: – \, 2 \, \left( \lambda_2 \,-\, \lambda_1 \right)\, u_1 \, u_2 \,.
\end{align}
\]

With

\[ u_1 = \cos \phi \,, \quad u_2 = \sin \phi
\]

it follows that

\[ \begin{align}
\alpha \:&=\: \lambda_2 * \sin^2 \phi \, + \, \lambda_1 * \cos^2 \phi \,, \\[8pt]
\gamma \:&=\: \lambda_2 * \cos^2 \phi \, + \, \lambda_1 * \sin^2 \phi \,, \\[8pt]
\beta \:&=\: – \, 2 \, \left( \lambda_2 \,-\, \lambda_1 \right)\, \cos \phi \, \sin \phi \,.
\end{align}
\]

These terms are intimately related to the geometrical data; expect them to play a major role in further considerations.

With the help of the coefficients of AE we can also show that det(Aq) > 0:

\[
\operatorname{det}\left( \pmb{\operatorname{A}}_q \right) \:=\: \left(\,\alpha \, \gamma \,-\, {1\over 4}\, \beta^2 \, \right) \:=\: { 1 \over \sigma_1^2 \, \sigma_2^2} \, \left(\, b\,c \,-\, a\,d \, \right)^2 \, \gt \, 0 \,.
\]

Thus Aq is an invertible matrix if AE is invertible. For standard conditions (σ1 >0, σ2 > 0) this is the case (see above). Furthermore, Aq is symmetric and thus its own transposed matrix.

Above we have got α, β, γ as some relatively simple functions of a, b, c, d. The inversion is not so trivial and we do not even try it here.

Instead we focus on how we can express σ1, σ2 and φ as functions of either (a, b, c, d) or (α, β, γ).

How to derive σ1, σ2 and φ from the coefficients of AE or Aq in the general case?

Let us assume we have (numerical) data for the coefficients of the quadratic form. Then we may want to calculate values for the length of the principal axes and the rotation angle φ of the corresponding ellipse. There are two ways to derive respective formulas:

  • Approach 1: Use trigonometric relations to directly solve the equation system.
  • Approach 2: Use an eigenvector decomposition of Aq.

Both ways are fun!

Direct derivation of σ1, σ2 and φ from Aq by using trigonometric relations

Trigonometric relations which we can use are:

\[ \begin{align}
\sin (2 \, \phi) \:&=\: 2 \,\cos (\phi)\, \sin(\phi) \,, \\[8pt]
\cos (2 \, \phi) \:&=\: 2 \,\cos^2 (\phi)\, -\, 1 \\[8pt]
\:&=\: 1 \,-\, 2\,\sin^2(\phi) \\[8pt]
\:&=\: \cos^2 (\phi)\, -\, \sin^2 (\phi) \,.
\end{align}
\]

Without loosing generality we assume

\[
\lambda_2 \:\ge \lambda_1 \,.
\]

In the end results would only differ by a rotation of π/2, if we had chosen otherwise.

By combining the above relations for the Aq-coefficients we find

\[ \begin{align}
\alpha \, + \, \gamma \,&=\, \lambda_1 \,+\, \lambda_2 \,, \\[8pt]
\gamma \,\, – \, \alpha \,&=\, \left( \, \lambda_2 \,-\, \lambda_1\,\right) \, \cos (2\,\phi) \,, \\[8pt]
\,-\, \beta \:&=\: \left( \, \lambda_2 \,-\, \lambda_1 \, \right)\, \sin (2\,\phi) \,.
\end{align}
\]

By squaring and adding the last two equations we further get

\[ \begin{align}
\lambda_2 \,+\, \lambda_1 \,&=\, \alpha \, + \gamma \,, \\[8pt]
\lambda_2 \,-\, \lambda_1 \,&=\, \left[\, \beta^2 \,+\, \left(\,\gamma \,-\,\alpha \,\right)^2\, \right]^{1/2} \,.
\end{align}
\]

Eventually we get:

\[ \begin{align}
\lambda_1 \,&=\, {1\over 2}\, \left[\, (\,\alpha\,+\,\gamma\,) \,-\, \left[\, \beta^2 \,+\, \left(\,\gamma \,-\,\alpha \,\right)^2\, \right]^{1/2} \, \right] \,, \\[8pt]
\lambda_2 \,&=\, {1\over 2}\, \left[\, (\,\alpha\,+\,\gamma\,) \,+\, \left[\, \beta^2 \,+\, \left(\,\gamma \,-\,\alpha \,\right)^2\, \right]^{1/2} \, \right] \,..
\end{align}
\]

For λ2 ≥ λ1 the rotation angle is given by

\[
\phi \:=\: \, {1 \over 2} \operatorname{arcsin}\left( {-\, \beta \over \left[ \beta^2 \, +\, \left( \gamma \,-\, \alpha \right)^2 \, \right]^{1/2} } \right)
\]

Example:

A typical example for a matrix would be one that appears in the context of a bivariate normal distribution with some correlation imposed onto the statistical variables:

\[ \pmb{\operatorname{A}}_q \:=\: {1\over 1\,-\, \rho^2}\,
\begin{pmatrix} 1 & -\,\rho \\ -\,\rho & 1 \end{pmatrix} \,\,,
\]

with ρ > 0 being the Pearson correlation coefficient. For this case we get for the half axes of a respective ellipse and the angle :

\[ \begin{align}
\sigma_1 \,& =\, \sqrt{\left(\, 1\,+\, \rho\,\right)\, } \,, \\[8pt]
\sigma_2 \,& =\, \sqrt{\left(\, 1\,-\, \rho\,\right)\, } \,, \\[8pt]
\phi \,& =\, \pi / 4 \,. \\[8pt]
\end{align}
\]

Direct derivation of σ1, σ2 and φ from AE by using trigonometric relations

We set

\[ \begin{align}
\epsilon_1 \,&=\, \sigma_1^2 \,=\, {1 \over \lambda_1 } \,, \\[8pt]
\epsilon_2 \,&=\, \sigma_2^2 \,=\, {1 \over \lambda_2 } \,.
\end{align}
\]

We have:

\[ \begin{align}
a^2 \,+\, b^2 \:&=\: \epsilon_1 * \cos^2 \phi \, + \, \epsilon_2 * \sin^2 \phi \,, \\[8pt]
c^2 \,+\, d^2 \:&=\: \epsilon_1 * \sin^2 \phi \, + \, \epsilon_2 * \cos^2 \phi \,, \\[8pt]
a\, c \,+\, b\, d \:&=\: \left( \, \epsilon_1 \,-\, \epsilon_2 \, \right) * \cos \phi * \sin \phi \,.
\end{align}
\]

This leads to

\[ \begin{align}
2 \,\left( \, a^2 \,+\, b^2 \, \right) \:&=\: \epsilon_1 * \left(\, 1 \,+\, \cos (2\phi) \, \right) \, + \, \epsilon_2 * \left( \, 1 \,-\, \cos (2\phi) \, \right) \,, \\[8pt]
2 \,\left( \, c^2 \,+\, d^2 \, \right) \:&=\: \epsilon_2 * \left(\, 1 \,+\, \cos (2\phi) \, \right) \, + \, \epsilon_1 * \left( \, 1 \,-\, \cos (2\phi)\, \right) \,, \\[8pt]
2 \, \left(\,a \,c \,+\, b\, d \,\right) \:&=\: \left(\, \epsilon_1 \,-\, \epsilon_2 \, \right) * \sin (2\phi) \,.
\end{align}
\]

We rearrange terms and get:

\[ \begin{align}
\epsilon_1 \,+\, \epsilon_2 \, +\, \left( \epsilon_1 \,-\, \epsilon_2 \right) * \cos (2\phi) \:&=\: \,+ 2 \, \left( \, a^2 \,+\, b^2 \, \right) \,, \\[8pt]
\,-\, \epsilon_1 \,-\, \epsilon_2 \,+\, \left( \epsilon_1 \,-\, \epsilon_2 \right) * \cos (2\phi) \:&=\: \,- 2 \, \left( \, c^2 \,+\, d^2 \, \right) \,, \\[8pt]
\left(\, \epsilon_1 \,-\, \epsilon_2 \, \right) * \sin (2\phi) \:&=\: \,+ 2 \, \left( \, a\,c \,+\, b\,d \, \right) \,.
\end{align}
\]

Let us define some further variables before we add and subtract the first two of the above equations:

\[ \begin{align}
r \:&=\: {1 \over 2} \left(\, a^2 \,+\, b^2 \,+\, c^2 \,+\, d^2 \, \right) \,, \\[8pt]
s_1 \:&=\: {1 \over 2} \left(\, a^2 \,+\, b^2 \,-\, c^2 \,-\, d^2 \, \right) \,, \\[8pt]
s_2 \:&=\: \left(\, a\,c \,+\, b\,d \, \right) \,, \\[8pt]
\pmb{s} \:&=\: \begin{pmatrix} s_1 \\ s_2 \end{pmatrix} \,, \\[8pt]
s \:&=\: \sqrt{ s_1^2 \,+\, s_2^2 } \, .
\end{align}
\]

Then adding two of the equations with the sin2φ and cos2φ above and using the third one results in:

\[
{1 \over 2} \, \left( \epsilon_1 \,-\, \epsilon_2 \right) \begin{pmatrix} \cos (2\phi) \\ \sin (2\phi) \end{pmatrix} \:=\: \begin{pmatrix} s_1 \\ s_2 \end{pmatrix}
\]

Taking the vector norm on both sides (with ε1 ≥ ε2) and adding two of the equations above results in:

\[ \begin{align}
\epsilon_1 \,\, – \,\, \epsilon_2 \:&=\: 2 * s \,, \\[8pt]
\epsilon_1 \, + \, \epsilon_2 \:&=\: 2 * r \,.
\end{align}
\]

This gives us:

\[ \begin{align}
\sigma_1^2 \,=\, \epsilon_1 \:&=\: r \,+\, s \,, \\[8pt]
\sigma_2^2 \,=\, \epsilon_2 \:&=\: r \,-\, s \,.
\end{align}
\]

In terms of the coefficients a, b, c, d:

\[ \begin{align}
\sigma_1^2 \,=\, {1 \over \lambda_1} \:&=\: {1 \over 2} \left[ \, a^2+b^2+c^2 +d^2 \,+\, \left[ 4 (ac + bd)^2 \, +\, \left( c^2+d^2 -a^2 -b^2\right)^2 \, \right]^{1/2} \right] \,, \\[8pt]
\sigma_2^2 \,=\, {1 \over \lambda_2} \:&=\: {1 \over 2} \left[ \, a^2+b^2+c^2 +d^2 \,-\, \left[ 4 (ac + bd)^2 \, +\, \left( c^2+d^2 -a^2 -b^2\right)^2 \, \right]^{1/2} \right] \,.
\end{align}
\]

Who said that life had to be easy?

But, it is relatively easy to prove:

\[
\sigma_1^2 * \sigma_2^2 \,=\, {1 \over \lambda_1^2 \, \lambda_2^2} \,=\, \left(\, a\,d\,-\, b\,c\ \, \right)^2 \,=\, \left[\operatorname{det}\left(\pmb{\operatorname{A}}_q \right)\right]^2 \,.
\]

So, we can indeed confirm :

\[ \begin{align}
\alpha \:&=\: { 1 \over \left(\,a\,d \,-\, b\,c\,\right)^2 } \, \left(\, c^2 \,+\, d^2 \,\right) \,, \\[8pt]
\gamma \:&=\: { 1 \over \left(\,a\,d \,-\, b\,c\,\right)^2 } \, \left(\, a^2 \,+\, b^2 \,\right) \,, \\[8pt]
\beta \:&=\: – \, 2\, { 1 \over \left(\,a\,d \,-\, b\,c\,\right)^2 } \, \left(\,a \,c \,+\, b \, d \,\right)\,.
\end{align}
\]

Determination of the inclination angle φ

For the determination of the angle φ we use:

\[
\begin{pmatrix} \operatorname{cos}(2\phi) \\ \operatorname{sin}(2\phi) \end{pmatrix} \:=\: {1 \over s} \begin{pmatrix} s_1 \\ s_2 \end{pmatrix} \,.
\]

We choose

\[
-\pi/2 \,\lt\, \phi \le \pi/2
\]

and get:

\[
\phi \:=\: {1 \over 2} \operatorname{arctan}\left({s_2 \over s_1}\right) \:=\: {1 \over 2} \, \operatorname{arctan}\left( { 2\left(ac \,+\, bd\right) \over (a^2 \,+\, b^2 \,-\, c^2 \,-\, d^2) } \right)
\]

Note: All in all there are four different solutions. The reason is that we alternatively could have requested λ2 ≥ λ1 and also chosen the angle π + φ. So, the ambiguity is due to a selection of the considered principal axis and rotational symmetries.

2nd way to a solution for σ1, σ2 and φ via eigendecomposition

For our second way of deriving formulas for σ1, σ2 and φ we use some linear algebra. This way is interesting for two reasons: It indicates how we can use the Python “linalg”-package together with Numpy to get results numerically. In addition we get familiar with a representation of the ellipse in a properly rotated CCS.

Above we have written down a symmetric matrix Aq describing an operation on the position vectors of points on our rotated ellipse:

\[
\pmb{v}_E^T \circ \pmb{\operatorname{A}}_q \circ \pmb{v}_E \:=\: 1 \,.
\]

We know from linear algebra that every symmetric matrix can be decomposed into a product of orthogonal matrices O, OT and a diagonal matrix. This reflects the so called eigendecomposition of a symmetric matrix. It is a unique decomposition in the sense that it has a uniquely defined solution in terms of the coefficients of the following matrices:

\[
\pmb{\operatorname{A}}_q \:=\: \pmb{\operatorname{O}} \circ \pmb{\operatorname{\Delta}}_{diag} \circ \pmb{\operatorname{O}}^T
\]

with

\[
\pmb{\operatorname{D}}_{diag} \:=\: \begin{pmatrix} \lambda_{u} & 0 \\ 0 & \lambda_{d} \end{pmatrix}
\]

The coefficients λu and λd are eigenvalues of both Ddiag and Aq.

Reason: Orthogonal matrices do not change eigenvalues of a transformed matrix. So, the diagonal elements of Ddiag are the eigenvalues of Aq. Linear algebra also tells us that the columns of the matrix O are given by the components of the normalized eigenvectors of Aq.

We can interpret O as a rotation matrix Rψ for some angle ψ:

\[
\pmb{v}_E^T \circ \pmb{\operatorname{A}}_q \circ \pmb{v}_E \:=\: \pmb{v}_E^T \circ
\pmb{\operatorname{R}}_{\psi} \circ \pmb{\operatorname{D}}_{diag} \circ \pmb{\operatorname{R}}_{\psi}^T \circ \pmb{v}_E \:=\: 1 \,.
\]

This means

\[
\left[ \pmb{\operatorname{R}}_{-\psi} \circ \pmb{v}_E \right]^T \circ
\pmb{\operatorname{D}}_{diag} \circ \left[ \pmb{\operatorname{R}}_{-\psi} \circ \pmb{v}_E \right] \:=\: 1 \,.
\]

The whole operation tells us a simple truth, which we are already familiar with. By our construction procedure for a rotated ellipse we know that a rotated CCS exists, in which the ellipse can be described as the result of a scaling operation (along the coordinate axes of the rotated CCS) applied to a unit circle. (This CCS is, of course, rotated by an angle φ against our working CCS in which the ellipse appears rotated.)

Above we had found

\[
\left[ \,\pmb{\operatorname{A}}_E^{-1} \circ \begin{pmatrix} x_E \\ y_E \end{pmatrix} \, \right]^T \, \bullet \,
\left[ \, \pmb{\operatorname{A}}_E^{-1} \circ \begin{pmatrix} x_E \\ y_E \end{pmatrix} \,\right] \:=\: 1
\]

With our matrices Rφ and scaling matrix DE we can rewrite this as

\[
\left[ \begin{pmatrix} x_E \\ y_E \end{pmatrix} \, \right]^T \, \circ \,
\left[ \pmb{\operatorname{D}}_E^{-1} \circ \pmb{\operatorname{R}}_{-\phi} \right]^T \bullet
\left[ \pmb{\operatorname{D}}_E^{-1} \circ \pmb{\operatorname{R}}_{-\phi} \right] \circ \begin{pmatrix} x_E \\ y_E \end{pmatrix} \:=\: 1 \,.
\]

So:

\[
\left[ \begin{pmatrix} x_E \\ y_E \end{pmatrix} \, \right]^T \, \circ \,
\left[ \pmb{\operatorname{R}}_{-\phi} \right]^T \circ \left[ \pmb{\operatorname{D}}_E^{-1} \right]^T \circ
\pmb{\operatorname{D}}_E^{-1} \circ \pmb{\operatorname{R}}_{-\phi} \circ \begin{pmatrix} x_E \\ y_E \end{pmatrix} \:=\: 1 \,.
\]

Remembering that a diagonal matrix is its own transposed matrix and that the inverse of aan orthogonal matrix (rotation) is its inverse we get:

\[
\left[ \begin{pmatrix} x_E \\ y_E \end{pmatrix} \, \right]^T \, \circ \,
\left[ \pmb{\operatorname{R}}_{\phi} \right] \circ \left[ \pmb{\operatorname{D}}_E^{-1} \circ
\pmb{\operatorname{D}}_E^{-1} \right] \circ \pmb{\operatorname{R}}_{\phi}^T \circ \begin{pmatrix} x_E \\ y_E \end{pmatrix} \:=\: 1 \,.
\]

Comparing expressions, we find

\[
\pmb{\operatorname{D}}_{diag} \:=\: \left[ \pmb{\operatorname{D}}_E^{-1} \circ
\pmb{\operatorname{D}}_E^{-1} \right] \,.
\]

And thus

\[ \begin{align}
\lambda_u \,=\, \lambda_1 \,&=\, {1 \over \sigma_1^2} \,, \\[8pt]
\lambda_d \,=\, \lambda_2 \,&=\, {1 \over \sigma_2^2} \,.
\end{align} \]

The eigenvalues of our symmetric matrix Aq are just λ1 and λ2.

Mathematically, a lengthy calculation (see below) will indeed reveal that the eigenvalues of a symmetric matrix Aq with coefficients α, 1/2*β and γ have the following form:

\[
\lambda_{1/2} \:=\: \lambda_{u/d} \:=\: {1 \over 2} \left[\, \left(\alpha \,+\, \gamma \right) \,\mp\, \left[ \beta^2 + \left(\gamma \,-\, \alpha \right)^2 \,\right]^{1/2} \, \right]
\]

This is, of course, exactly what we have found some minutes ago by directly solving the equations with the trigonometric terms.

We will prove the fact that these indeed are valid eigenvalues in a minute. Let us first look at respective eigenvectors ξ1/2. To get them we must solve the equations resulting from

\[
\left( \begin{pmatrix} \alpha & \beta / 2 \\ \beta / 2 & \gamma \end{pmatrix} \,-\, \begin{pmatrix} \lambda_{1/2} & 0 \\ 0 & \lambda_{1/2} \end{pmatrix} \right) \,\circ \, \pmb{\xi_{1/2}} \:=\: \pmb{0},
\]

with

\[
\pmb{\xi_1} \,=\, \begin{pmatrix} \xi_{1,x} \\ \xi_{1,y} \end{pmatrix}, \quad \pmb{\xi_2} \,=\, \begin{pmatrix} \xi_{2,x} \\ \xi_{2,y} \end{pmatrix}
\]

Below we will show that the following vectors fulfill the conditions (up to a common factor in the components):

\[ \begin{align}
\lambda_1 \: &: \quad \pmb{\xi_1} \:=\: \left(\, {1 \over \beta} \left( (\alpha \,-\, \gamma) \,-\, \left[\, \beta^2 \,+\, \left(\gamma \,-\, \alpha \right)^2\,\right]^{1/2} \right), \: 1 \, \right)^T \,, \\[8pt]
\lambda_2 \: &: \quad \pmb{\xi_2} \:=\: \left(\, {1 \over \beta} \left( (\alpha \,-\, \gamma) \,+\, \left[\, \beta^2 \,+\, \left(\gamma \,-\, \alpha \right)^2\,\right]^{1/2} \right), \: 1 \, \right)^T \,.
\end{align}
\]

for the eigenvalues

\[ \begin{align}
\lambda_1 \:&=\: {1 \over 2} \left(\, \left(\alpha \,+\, \gamma \right) \,-\, \left[ \beta^2 \,+\, \left(\gamma \,-\, \alpha \right)^2 \,\right]^{1/2} \, \right) \,, \\[8pt]
\lambda_2 \:&=\: {1 \over 2} \left(\, \left(\alpha \,+\, \gamma \right) \,+\, \left[ \beta^2 \,+\, \left(\gamma \,-\, \alpha \right)^2 \,\right]^{1/2} \, \right) \,.
\end{align}
\]

As usual, the T at the formulas for the vectors symbolizes a transposition operation.

Note that the vector components given above are not normalized. This is important for performing numerical checks as Numpy and linear algebra programs would typically give you normalized eigenvectors with a length = 1. But you can easily compensate for this by working with

\[ \begin{align}
\lambda_1 \: &: \quad \pmb{\xi_1^n} \:=\: {1 \over \|\pmb{\xi_1}\|}\, \pmb{\xi_1} \,, \\[8pt]
\lambda_2 \: &: \quad \pmb{\xi_2^n} \:=\: {1 \over \|\pmb{\xi_2}\|}\, \pmb{\xi_2} \,.
\end{align}
\]

Proof for the eigenvalues and eigenvector components

We just prove that the eigenvector conditions are e.g. fulfilled for the components of the first eigenvector ξ1 and λ1 = λu.

\[ \begin{align}
\left(\alpha \,-\, \lambda_1 \right) * \xi_{1,x} \,+\, {1 \over 2} \beta * \xi_{1,y} \,&=\, 0 \\[8pt]
{1 \over 2} \beta * \xi_{1,x} \,+\, \left( \gamma \,-\, \lambda_1 \right) * \xi_{2,y} \,&=\, 0
\end{align}
\]

(The steps for the second eigenvector are completely analogous).

We start with the condition for the first component:

\[ \begin{align}
&\left( \alpha \,-\,
{1\over 2}\left[\,\left(\alpha \, + \, \gamma\right) \,-\, \left[ \beta^2 \,+\, \left( \alpha \,-\, \gamma \right)^2 \right]^{1/2} \right] \right) * \\[8pt]
& {1 \over \beta}\,
\left[\, \left(\alpha \,-\, \gamma\right) \,-\, \left[ \beta^2 \,+\, \left(\alpha \,-\, \gamma \right)^2 \right]^{1/2} \,\right] \,+\, {\beta \over 2 }
\,=\, 0
\end{align}
\]
\[ \begin{align}
& {1 \over 2 } \left[ \left(\alpha \,-\,\gamma\right) \,+\, \left[ \beta^2 \,+\, \left( \alpha \,-\, \gamma \right)^2 \right]^{1/2} \right] * \\[8pt]
& {1 \over \beta}\,
\left[\, \left(\alpha \,-\, \gamma\right) \,-\, \left[ \beta^2 \,+\, \left(\alpha \,-\, \gamma \right)^2 \right]^{1/2} \,\right] \,+\, {\beta \over 2 }
\,=\, 0 \,.
\end{align}
\]
\[
{1 \over 2 \, \beta} \left[ (\alpha \,-\,\gamma)^2 \,-\, \beta^2 \,-\, (\alpha \,-\,\gamma)^2 \right] \,+\, {\beta \over 2 } \,=\, 0
\]

The last relation is obviously true.

You can perform a similar calculation for the other eigenvector component:

\[ \begin{align}
{1 \over 2} \, \beta & {1 \over \beta}\,
\left[\, \left(\alpha \,-\, \gamma\right) \,-\, \left[ \beta^2 \,+\, \left(\alpha \,-\, \gamma \right)^2 \right]^{1/2} \,\right] \,+\, \\
&
\left( \gamma \, -\,
{1\over 2}\left[\,\left(\alpha \, + \, \gamma\right) \,-\, \left[ \beta^2 \,+\, \left( \alpha \,-\, \gamma \right)^2 \right]^{1/2} \right] \right) * 1 \,=\, 0
\end{align}
\]

Thus:

\[ \begin{align}
&{1 \over 2} \, \left(\alpha \,-\, \gamma\right) \,-\, {1 \over 2} \left[ \beta^2 \,+\, \left(\alpha \,-\, \gamma \right)^2 \right]^{1/2} \\
-\, &{1\over 2}\left(\alpha \,-\, \gamma\right) \,+\, {1\over 2}\left[ \beta^2 \,+\, \left( \alpha \,-\, \gamma \right)^2 \right]^{1/2} \,=\, 0
\end{align}
\]

True, again.

In a very similar exercise one can show that the scalar product of the eigenvectors is equal to zero:

\[ \begin{align}
& {1 \over \beta^2}\,
\left[\, \left(\alpha \,-\, \gamma\right)^2 \,-\, \beta^2 \,+\, \left(\alpha \,-\, \gamma \right)^2 /\,\right] \,+\, 1 \\[8pt]
& – {1 \over \beta^2} * \beta^2 \,*\,1 \,=\, 0 \,.
\end{align}
\]

I.e. andindeed

\[
\pmb{\xi_1} \bullet \pmb{\xi_1} \,=\, \left( \xi_{1,x}, \, \xi_{1,y} \right) \circ \begin{pmatrix} \xi_{2,x} \\ \xi_{2,y} \end{pmatrix} \,= \, 0 \,.
\]

The eigenvectors are perpendicular to each other. Exactly, what we expect for the orientations of the principal axes of an ellipse against each other.

Rotation angle from coefficients of Aq

We still need a formula for the rotation angle(s). From linear algebra results related to an eigendecomposition we know that the orthogonal (rotation) matrices consist of columns of the normalized eigenvectors. With the components given in terms of our un-rotated CCS, in which we basically work. These vectors point along the principal axes of our ellipse. Thus the components of these eigenvectors define our aspired rotation angles of the ellipse’s principal axes against the x-axis of our CCS.

Let us prove this. By assuming

\[ \begin{align}
\cos (\phi_1) \,&=\, \xi_{1,x}^n \,, \\[8pt]
\sin (\phi_1) \,&=\, \xi_{1,y}^n
\end{align}
\]

and using

\[
\sin(2\phi_1) \,=\, 2\, \sin (\phi_1) \, \cos (\phi_1) \,,
\]

we get:

\[ \begin{align}
\sin (2 \phi_1) \,=\,
2 * { \xi_{1,x} * \xi_{1,y} \over \left[\, \xi_{1,x}^2 \, + \, \xi_{1,y}^2 \,\right] } \,.
\end{align}
\]

Thus

\[ \begin{align}
\operatorname{sin}(2 \phi_1) \,&=\,
2 \,\, { {1 \over \large{\beta}} \left( (\alpha \,-\, \gamma) \,-\, \left[\, \beta^2 \,+\, \left(\gamma \,-\, \alpha \right)^2\,\right]^{1/2} \right) \,*\, 1
\over
\left[\, \left( {1 \over \large{\beta}} \left( (\alpha \,-\, \gamma) \,-\, \left[\, \beta^2 \,+\, \left(\gamma \,-\, \alpha \right)^2\,\right]^{1/2} \right) \right)^2
\,+\, 1^2 \right] } \\[8pt]
&=\, 2\,\, { {1 \over \large{\beta}} \left( t \,-\, z \right)
\over
{1 \over \large{\beta}^2 \phantom{\large{]}} } \left[\, \beta^2 \,+\, \left(\, t \,-\, z \,\right)^2 \right] }
\end{align}
\]

with

\[ \begin{align}
t \,&=\, (\alpha \,-\, \gamma) \,, \\[8pt]
z \,&=\, \left[\, \beta^2 \,+\, \left(\gamma \,-\, \alpha \right)^2\,\right]^{1/2} \,.
\end{align}
\]

This looks very differently from the simple expression we got above. And a direct approach is cumbersome. The trick is to multiply nominator and denominator by a convenience factor

\[
\left( t \,+\, z \right),
\]

and exploit

\[ \begin{align}
\left( t \,-\, z \right) \, \left( t \,+\, z \right) \,&=\, t^2 \,-\, z^2 \,, \\[8pt]
\left( t \,-\, z \right) \, \left( t \,+\, z \right) \,&=\,\, – \,\beta^2
\end{align}
\]

to get

\[ \begin{align}
&2 * \beta \, { (t\,-\, z) * ( t\,+\, z) \over \left[ \beta^2 \, + \,( t \,-\, z )^2 \right] * (t \,+\, z) } \\[8pt]
&=\: 2 * \beta \, { – \beta^2 \over \beta^2 (t\,+\,z) \,-\, \beta^2 (t\,-\,z) } \\
&=\: – \, {\beta \over \left[\, \beta^2 \,+\, (\alpha \,-\, \gamma)^2 \,\right]^{1/2} }
\end{align}
\]

This means that our 2nd approach gives us the result

\[
\operatorname{sin}\, (2 \phi_1) \:=\: \, – \, { \beta \over \left[\, \beta^2 \,+\, (\alpha \,-\, \gamma)^2 \,\right]^{1/2} }\,,
\]

which is of course identical to the result we got with our first solution approach. It is clear that the second axis has an inclination by φ +- π / 2:

\[
\phi_2\, =\, \phi_1 \,\pm\, \pi/2.
\]

In general the angles have a natural ambiguity of π.

Conclusion

In this post I have shown how one can derive essential properties of centered, but rotated ellipses from matrix-based representations. Such calculations become relevant when e.g. experimental or numerical data only deliver the coefficients of a quadratic form for the ellipse.

We have first established the relation of the coefficients of a matrix that defines an ellipse by a combined scaling and rotation operation with the coefficients of a matrix which defines an ellipse as a quadratic form of the components of position vectors. In addition we have shown how the coefficients of both matrices are related to quantities like the lengths of the principal axes of the ellipse and the inclination of these axes against the x-axis of the Cartesian coordinate system in which the ellipse is described via position vectors. So, if one of the matrices is given we can numerically calculate the ellipse’s main properties.

In the next post of this mini-series

Properties of ellipses by matrix coefficients – II – coordinates of points with extremal y-values

we have a look at the x- and y-coordinates of points on an ellipse with extremal y-values. All in terms of the matrix coefficients we are now familiar with.