9.48 min read

Immediate Uses of SVD

From the SVD $A = U\Sigma V^\top$ , several fundamental properties of $A$ can be read off immediately. The rank of $A$ is the number of nonzero singular values. The column space is spanned by the left singular vectors $\mathbf{u}_i$ corresponding to nonzero $\sigma_i$ . The null space is spanned by the right singular vectors $\mathbf{v}_i$ corresponding to zero $\sigma_i$ .

The pseudoinverse $A^+ = V \Sigma^+ U^\top$ (where $\Sigma^+$ replaces each nonzero $\sigma_i$ with $1/\sigma_i$ ) gives the minimum-norm least-squares solution. The condition number $\sigma_1 / \sigma_r$ measures numerical sensitivity.

Formal View

Corollary 9.1 — Properties from SVD

Let

A = U\Sigma V^\top

with

r = \operatorname{rank}(A)

. Then: -

\operatorname{rank}(A) = r = \#\{\sigma_i > 0\}

\operatorname{col}(A) = \operatorname{span}(\mathbf{u}_1, \ldots, \mathbf{u}_r)

\ker(A) = \operatorname{span}(\mathbf{v}_{r+1}, \ldots, \mathbf{v}_n)

A^+ = V\Sigma^+ U^\top

(pseudoinverse)

Why This Matters

The SVD gives the most numerically stable way to compute rank, null spaces, and pseudoinverses.

Numerical rank computation: software like MATLAB uses SVD (not determinant) to reliably determine rank.
Pseudoinverse for over/under-determined systems handles all cases.
Condition number $\sigma_1/\sigma_r$ alerts to near-singular matrices before solving systems.

Learning Resources

SVD applications: rank, null space, pseudoinverse

MIT OpenCourseWare

Strang shows how SVD reveals the four fundamental subspaces and pseudoinverse.

45 min

Matrix pseudoinverse

Steve Brunton

Computing and using the Moore-Penrose pseudoinverse via SVD.

15 min

Quiz

Question 1

For $A = U\Sigma V^\top$ with singular values $\{3, 1, 0\}$ , what is $\operatorname{rank}(A)$ ?

Question 2

The null space of $A$ is spanned by the right singular vectors corresponding to zero singular values.

Common Mistakes

Confusing left and right singular vectors — left ( $U$ ) spans column space; right ( $V$ ) spans null space for zero singular values.
Forgetting that the pseudoinverse $A^+$ is not the same as $A^{-1}$ .