Derivation — Constructing U from V via the Spectral Theorem

The previous note built

V

and

\Sigma

from the spectral theorem applied to

A^\top A

, but deferred

U

's construction. This sub-note completes it — handling carefully the one genuine subtlety LA-I's preview glossed over: what happens when some singular value is exactly zero, so

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

is undefined.

Learning objectives. By the end of this note you should be able to:

construct $U$ 's columns for nonzero singular values and verify their orthonormality;
explain why the zero-singular-value case needs separate handling;
complete $U$ to a full orthonormal basis when $\text{rank}(A)<\min(m,n)$ ;
verify the full SVD relation $AV=U\Sigma$ from the construction.

In this lesson

Constructing $U$ for nonzero singular values
Handling zero singular values
Verifying the full construction

Before you start

Existence of the SVD for Every Matrix

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