Rank-one update singularity criterion ($A+u{v}^T\text{ singular}⟺1+v^T{A}^{-1}u=0$)

For an invertible matrix $A$, the rank-one matrix update $B=A+u{v}^T$ is a singular matrix exactly when $1+v^T{A}^{-1}u=0$. This follows directly from the matrix determinant lemma. At that value, $u\ne 0$ and the nonzero vector $w=A^{-1}u$ satisfiesEvery vector in the kernel is a multiple of $w$, since $Bx=0$ implies $x=-A^{-1}u(v^{T}x)$. The matrix rank therefore drops by exactly one.

For $B(t)=I+tu{v}^T$ with $u=(1,2{)}^T$ and $v=(3,-1{)}^T$, the update becomes a singular matrix at $t=-1$, and $B(t)u=(1+t)u$. At $t=-0.99$, this direction is compressed by a factor $0.01$ and the inverse amplifies it by a factor 100. The actual spectral norm of the matrix inverse, which maximizes amplification over every direction, is about $700.17853409$.

The exact formula can expose a large condition number in the updated linear system even when the original one had a small condition number. It does not guarantee numerical stability merely because its evaluation uses fewer arithmetic operations. The denominator condition certifies existence; the condition number and floating-point behavior address different questions about using the resulting answer.

Let $a,b,c\in {\mathbb{R}}^n$, with $c\ne 0$. The equation $x=a+(b\cdot x)c$ is the linear system $(I-c{b}^T)x=a$, a rank-one matrix update of the identity matrix. Every solution has $x=a+\lambda c$. Substitution reduces the entire problem toIf $b\cdot c\ne 1$, there is one solution, $x=a+c(b\cdot a)/(1-b\cdot c)$. If $b\cdot c=1$ and $b\cdot a=0$, every $\lambda \in \mathbb{R}$ works and the solution set is the straight line $a+\mathbb{R}c$. If $b\cdot c=1$ and $b\cdot a\ne 0$, there is no solution.

The first reduction follows directly from the original equation in any dimension. In three dimensions it can also be obtained by taking the cross product with $c$, giving $(x-a)\times c=0$. The singular denominator therefore represents two different possibilities, inconsistency or a family of solutions, and must not simply be discarded as an undefined formula.