Rank-One Matrix
Chapter 4: QR Decomposition and Numerical Rank — Rank-1 Perturbations (also appears in Ch. 5, Ch. 8)
From the book
Chapter 4: QR Decomposition and Numerical Rank. In the chapter mind map this icon labels Rank-1 Perturbations:($I + \mathbf{u}\mathbf{v}^\top$). The discussion below is excerpted and lightly edited from § Rank: Advanced Properties and Numerical Computation in Mathematics for AI and Machine Learning. Related material also appears in Chapter 5 (Inverses& Woodbury Identity), Chapter 8 (Outer Product Expansion: $A = \sum \sigma_i \mathbf{u}_i\mathbf{v}_i^\top$).
The basic definition of rank and linear independence were introduced in the matrix chapter. The Rank-Nullity Theorem and the relationship between rank and the dimensions of the four fundamental subspaces are covered in the matrix chapter. Here we focus on advanced properties and numerical methods for computing rank.
What this drawing shows
What you see. Represents a matrix formed as one outer product, mapping inputs through a single direction of variation.
In the mind map. Chapter 4 — Rank-1 Perturbations:(). See From the book above for definitions, figures, and worked examples.
Also appears in Ch. 5 (Inverses& Woodbury Identity); Ch. 8 (Outer Product Expansion).
Where to read next
Read the full definitions, figures, and worked examples in Chapter 4: QR Decomposition and Numerical Rank — see the mind-map node Rank-1 Perturbations:().
This concept is also referenced in Chapter 5 (Inverses& Woodbury Identity); Chapter 8 (Outer Product Expansion).