Recipe

Assess a matrix's rank and conditioning from its singular values

Use when: you have $A$ 's singular values (or its SVD) and need its rank, how trustworthy that rank is numerically, or how sensitive $A\mathbf x=\mathbf b$ is to input error

Goal: the exact rank, the numerical rank at a chosen tolerance, and the condition number $\kappa(A)$ , with what each one means for solving with $A$

Taught in Rank, numerical rank, and the condition number

Quick recipe

List the singular values and check which are exactly zero
Pick a tolerance and count numerical rank against it
Compute the condition number from the exact smallest and largest singular values
Use $\kappa(A)$ to bound the solution's own relative error

1
Write $\sigma_1\ge\cdots\ge\sigma_n$ from the SVD; the exact rank is how many of them are strictly greater than zero, no matter how small.

Example

$A$ has singular values $5, 0.05, 0.005$ from largest to smallest; none is exactly zero, so the exact rank is $3$ .

Why: exact rank is a strict positive-or-zero test on each singular value, the same test the SVD's own rank factorization already relies on.
2
Choose a threshold $\epsilon$ suited to the data's own precision, then count how many singular values exceed $\epsilon$ ; that count is the numerical rank.

Example

With $\epsilon=0.01$ : $5>0.01$ and $0.05>0.01$ both count, while $0.005<0.01$ does not, giving numerical rank $2$ .
- a singular value sits comfortably above $\epsilon$ — count it as genuine structure.
- a singular value sits at or below $\epsilon$ — drop it, treating it as rounding noise rather than real structure.
Why: numerical rank is what a computation should actually trust as real structure once floating-point noise is accounted for.
3
Divide the largest singular value by the smallest one that is still strictly nonzero: $\kappa(A)=\sigma_1/\sigma_r$ , using the exact rank's own $r$ , not the numerical-rank cutoff.

Example

$\kappa(A)=5/0.005=1000$ .
- $A$ has no exactly-zero singular value — $\kappa(A)$ is a genuine finite number.
- $A$ has at least one exactly-zero singular value — $\kappa(A)$ is treated as infinite, signaling $A$ is not invertible at all.
Why: this measures the true worst-case sensitivity built into $A$ itself, independent of whatever tolerance the numerical-rank calculation happened to use.
4
Multiply $\kappa(A)$ by the relative error already present in $\mathbf b$ to get the worst-case relative error that could appear in the solution $\mathbf x$ .

Example

If $\mathbf b$ carries a $0.1\%$ relative error, the bound is $1000\times0.001=1$ , meaning the solution's error could be as large as the solution itself.

Why: this bound is what makes a large condition number a genuine numerical hazard, warning that a tiny input error can swallow the whole answer.

Common mistakes

Confusing numerical rank's tolerance-dependent count with exact rank's strict positive-or-zero test.
Computing $\kappa(A)$ using the numerical rank's smallest surviving singular value instead of the true smallest nonzero one.
Treating a large $\kappa(A)$ as impossible just because $A$ is technically invertible; the two facts are unrelated.
Picking a tolerance $\epsilon$ with no connection to the data's own measurement precision.