2D SVD Visualization

For any \(2 \times 2\) matrix \(A\), the singular value decomposition is the factorization

\[ A \;=\; U\,\Sigma\,V^{T}, \]

which expresses the linear map \(\mathbf{x} \mapsto A\mathbf{x}\) as three geometric steps performed right-to-left.

Components:

• \(V^{T}\) is a \(2 \times 2\) orthogonal matrix — a rotation (or rotation + reflection) of the input plane onto the principal input axes.
• \(\Sigma = \operatorname{diag}(\sigma_1, \sigma_2)\) with \(\sigma_1 \geq \sigma_2 \geq 0\) — a scaling along those axes that turns the unit circle into an ellipse with semi-axes \(\sigma_1\) and \(\sigma_2\).
• \(U\) is a \(2 \times 2\) orthogonal matrix — a final rotation that places the ellipse in the output plane.

Geometric reading of this visualizer:

• The unit circle is mapped to an ellipse whose semi-axis lengths are the singular values \(\sigma_1, \sigma_2\).
• The right singular vectors (columns of \(V\)) are the pre-images of the ellipse's principal axes; the left singular vectors (columns of \(U\)) are those principal axes themselves.
• \(\sigma_2 = 0\) means \(A\) is rank-deficient: the ellipse collapses to a line segment.

For step-by-step numerical SVD with surd / fraction output of \(U\), \(\Sigma\), \(V\) for any \(m \times n\) matrix, see the Singular Value Decomposition (SVD) Calculator.