SVD Computation

Understanding the elementwise construction of $y = U \Sigma V^* x$

Formulation

y = U \Sigma V^* x

By breaking down the matrix multiplication into its core components, we can see exactly how the Singular Value Decomposition maps the input vector $x$ to the output vector $y$ .

y_i = \sum_{j} U_{ij} \sigma_j (V^*_{j,:} \cdot x)

Notice that each entry $y_i$ is constructed by taking the dot product of the input with the rows of $V^*$ , scaling by the singular values $\sigma$ , and recombining using the columns (or rows) of $U$ .

Instructions

Hover over or click the elements in the grid to see how they interact.
Select an output index $i$ to see how $y_i$ is computed.
Select a component index $j$ to isolate one singular value path.

Target Output Element (i)

Singular Value Path (j)

y_1

y_2

y_3

U_{11}

U_{21}

U_{31}

U_{12}

U_{22}

U_{32}

U_{13}

U_{23}

U_{33}

\sigma_1

0

0

0

\sigma_2

0

0

0

\sigma_3

V^*_{11}

V^*_{12}

V^*_{13}

V^*_{21}

V^*_{22}

V^*_{23}

V^*_{31}

V^*_{32}

V^*_{33}

x_1

x_2

x_3

Active Computation

y = \sum_{j=1}^3 \mathbf{u}_j \sigma_j (\mathbf{v}_j^* x)