How does a small perturbation to a column change the exact rank of a matrix?
Conditions
- The original columns are linearly dependent.
- A perturbation adds a nonzero component to a column in a new dimension.
Reasoning, step by step
- Start with a matrix whose columns are dependent (e.g., confined to a plane).
- Add a nonzero z-component to the third column.
- Observe that the columns now span the full three-dimensional space.
- Conclude that the exact rank increases to three (full rank).
- Note that if the z-component is tiny, the columns are nearly dependent despite the full exact rank.
Example
Giving the third column a nonzero z-component makes the example full rank. If that component is tiny, however, the columns remain nearly dependent and calculations can be sensitive.
Common misconceptions
- Believing that a tiny perturbation does not change the exact rank.
- Confusing exact rank with numerical stability or conditioning.
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