 Solution

Use the singular value decomposition and the orthogonal invariance of the Euclidean norm. Put $z=V^{T}x$ and $c=U^{T}b$. Since precisely ${\sigma }_1,\dots ,{\sigma }_r$ are positive,The second sum is independent of $x$ and cannot be removed. The first sum vanishes exactly when $z_j=c_j/{\sigma }_j$ for $1\le j\le r$, while $z_{r+1},\dots ,z_n$ are arbitrary. Thus every least-squares solution isThe unavoidable linear-system residual is the component of $b$ orthogonal to the column space of $A$:The minimum-norm least-squares solution sets all ${\alpha }_j=0$ and equals $A^{+}b$, where $A^{+}$ is the Moore-Penrose inverse. The squared minimum, rather than the minimum itself, is the second sum in the first display.