Visualizing Matrices

There are various scenarios (e.g., covariance matrices, inertia matrices, quadratic forms) in which you'd want to represent a (square) matrix visually, and ultimately it comes down to Eigen decompositions.

In graphics, it's generally preferable to draw an "unrotated" version of your graphic, and then apply a rotation. Thus, the methods below will focus on extracting semi-major axis lengths and a rotation from the Eigen decomposition. Example code will be given in Matlab.

For testing, here's some Matlab code for generating a random \(n\times n\) positive-definite matrix:

function A = generateSPDmatrix(n)
% Generate a dense n x n symmetric, positive definite matrix

A = rand(n,n); % generate a random n x n matrix

% construct a symmetric matrix using either
A = 0.5*(A+A'); OR
A = A*A';
% The first is significantly faster: O(n^2) compared to O(n^3)

% since A(i,j) < 1 by construction and a symmetric diagonally dominant matrix
%   is symmetric positive definite, which can be ensured by adding nI
A = A + n*eye(n);

end

Two/Three-Dimensional Positive Definite Matrix (Ellipse/Ellipsoid)

Matrix: \(\boldsymbol A\in \mathbb{R}^{2\times 2}\) or \(\boldsymbol A\in \mathbb{R}^{3\times 3}\), \(\boldsymbol A > 0\)

Get the principal axis lengths:

• Find eigenvalues of \(\boldsymbol A\). These are the (ordered) semi-major axis lengths.
  • Let's say that by convention, \(\boldsymbol A\) is expressed in frame \(W\), and when it's expressed in frame \(F\), it's diagonal (\(\boldsymbol D\)) with the eigenvalues on the diagonal.

Get the rotation matrix:

• Find the (ordered) eigenvectors of \(\boldsymbol A\). These form the column vectors of \(\boldsymbol R_F^W\).
• Check the determinant of \(\boldsymbol R_F^W\); if it's -1, then flip the sign of the last column to make the determinant +1.
• From matrix basis change rules, you can check your work by making sure that \(\boldsymbol D=\left(\boldsymbol R_F^W\right)^{-1}\boldsymbol A\boldsymbol R_F^W\).

Matlab code:

% Get random positive definite matrix
A = generateRandom2x2PDMatrix();

% Extract axis lengths and rotation
[R,D] = eig(A);
x_axis_len = D(1,1);
y_axis_len = D(2,2);
if det(R) < 0
    R(:,2) = -1 * R(:,2);
end

% Draw unrotated ellipse
theta = linspace(0,2*pi,100);
coords = [x_axis_len * cos(theta); y_axis_len * sin(theta)];
plot(coords(1,:),coords(2,:),'k--')
hold on; grid on

% Draw rotated ellipse (R acts like active rotation
% since it's B2W convention)
rotated_coords = R * coords;
plot(rotated_coords(1,:),rotated_coords(2,:),'k-','Linewidth',2.0)
hold off

A

function A = generateRandom2x2PDMatrix()
A = rand(2,2);
A = A*A';
A = A + 2*eye(2);
end

$$\boldsymbol A=\begin{bmatrix}3.0114 & 0.9353 \\ 0.9353 & 2.9723\end{bmatrix}$$