SVD vs Eigenvalue

Comparing two fundamental matrix perspectives

Formulation

A = U \Sigma V^*

A\mathbf{v}_i = \sigma_i \mathbf{u}_i

The SVD views the matrix as mapping a vector from a domain space to a codomain space. It identifies an orthogonal basis $V$ in the domain that maps exactly to an orthogonal basis $U$ in the codomain, scaled by $\Sigma$ .

Geometric Model

A = \begin{bmatrix} 2 & 1 \\ 0 & 1 \end{bmatrix}

A non-symmetric 2x2 matrix

The domain is drawn as an inset circle.
Orthogonal vectors $\mathbf{v}_1, \mathbf{v}_2$ form a clean grid in the domain.
They map to orthogonal vectors $\sigma_1\mathbf{u}_1, \sigma_2\mathbf{u}_2$ which form the major/minor axes of the resulting ellipse.

Controls

θInput Vector

\mathbf{x}

45°