This chapter starts with the simplest square full-rank case, where A⁻¹ exists, and then extends the same idea to tall, wide and rank-deficient matrices. The pseudoinverse A⁺ recovers the reversible part of a transformation: it maps col(A) back to row(A), sends the left null space to 0⃗ and leads to the projection operators A⁺A and AA⁺.
A true inverse exists only when the transformation is both one-to-one and onto. In all other cases, the pseudoinverse reverses the part of the transformation that remains meaningful.
The chapter ends with a summary table and a concrete geometric example of A: ℝ³ → ℝ² and A⁺: ℝ² → ℝ³.
It does not create a true inverse when one does not exist. Instead, it reverses the reachable part of the map: vectors in col(A) are sent back to row(A), while vectors in ℓ-null(A) are sent to 0⃗.
The PDF is a single document. The page links below are best-effort: most browsers support them, but some viewers may ignore the page hint.
| Topic | Pages |
|---|---|
| Simplest case: square full-rank A | 1 |
| Tall full-column-rank A | 2–4 |
| Wide full-row-rank A | 5–7 |
| General rank-deficient | 8–12 |
| Summary table | 13 |
| Example of A: ℝ³ → ℝ², A⁺: ℝ² → ℝ³ | 13–16 |
Because they separate what is preserved from what is lost. A⁺A projects domain vectors onto row(A), while AA⁺ projects codomain vectors onto col(A).