Rank
Definition
rank(A) = dimension of column space = dimension of row space. Computed by counting pivot positions in row echelon form.
Examples
Example 1. Rank of [[1,2,3],[2,4,6]].
Solution. Row 2 = 2·Row 1. Rank = 1.
In Depth
The rank of a matrix \(A\) is the dimension of its column space (equivalently, row space). It equals the number of pivots in the row echelon form. The rank-nullity theorem: \(\text{rank}(A)+\text{nullity}(A)=n\) (number of columns). Row rank equals column rank — a non-obvious fact.
A matrix has full column rank if its columns are linearly independent (rank = number of columns); full row rank if its rows are linearly independent (rank = number of rows). A square matrix is invertible iff it has full rank iff its determinant is nonzero.
Rank and linear systems: \(A\mathbf{x}=\mathbf{b}\) is consistent iff \(\text{rank}(A)=\text{rank}([A|\mathbf{b}])\). If consistent, the solution is unique iff \(A\) has full column rank; otherwise there are infinitely many solutions parameterized by the null space.
The rank of a product: \(\text{rank}(AB)\leq\min(\text{rank}(A),\text{rank}(B))\). Multiplying by an invertible matrix preserves rank. The rank of \(A+B\) satisfies \(|\text{rank}(A)-\text{rank}(B)|\leq\text{rank}(A+B)\leq\text{rank}(A)+\text{rank}(B)\).
Low-rank approximation: the best rank-\(k\) approximation to \(A\) (in Frobenius or spectral norm) is given by the truncated SVD \(A_k=\sum_{i=1}^k\sigma_i\mathbf{u}_i\mathbf{v}_i^T\). This is the mathematical basis of image compression, collaborative filtering, and latent semantic analysis.
Key Properties & Applications
Matrix rank is computed numerically using the SVD: the rank equals the number of singular values above a threshold (accounting for floating-point errors). The numerical rank (rank with tolerance \(\epsilon\)) is more practically useful than the exact rank for noisy data.
The rank of a matrix product: \(\text{rank}(AB)\leq\min(\text{rank}(A),\text{rank}(B))\). For square invertible \(C\): \(\text{rank}(CA)=\text{rank}(A)\). The rank of \(A^TA\) equals the rank of \(A\) (since \(\text{Null}(A^TA)=\text{Null}(A)\)).
Low-rank structure is ubiquitous in applications. Recommendation systems model user-item interactions as a low-rank matrix (collaborative filtering). Image compression uses low-rank approximations. Quantum state tomography reconstructs low-rank density matrices from measurements.
Further Reading & Context
The study of rank connects to many areas of mathematics and its applications. Understanding the foundational definitions and theorems provides the basis for advanced work in analysis, algebra, and applied mathematics.
Historical development: most mathematical concepts evolved over centuries, with contributions from mathematicians across many cultures. The modern axiomatic treatment provides rigor, while computational tools enable practical application.
In modern mathematics, this topic appears in graduate courses and research across pure and applied mathematics. Connections to computer science, physics, and engineering make it a versatile and important area of study. Mastery of the core results and techniques opens doors to research in number theory, analysis, geometry, and beyond.
Recommended next steps: work through the standard theorems with full proofs, explore the connections to related topics listed above, and practice with a variety of problems ranging from computational exercises to theoretical proofs. The interplay between different areas of mathematics is one of the subject's greatest rewards.