rankA

rankA is a term sometimes used to denote the rank of a matrix A. The rank of A is a fundamental concept in linear algebra: it is the maximum number of linearly independent rows or columns of A and equals the dimension of both the row space and the column space.

In standard notation, the rank is written as rank(A). Some texts or software may refer to rankA(A) or rankA as a shorthand for rank(A). The concept applies to matrices over any field and to linear maps represented by matrices.

Key properties and relationships:

- For A in F^(m×n), rank(A) lies between 0 and min(m,n).

- Invariance: rank(PAQ) = rank(A) for any invertible matrices P and Q.

- Rank-nullity relation: rank(A) + nullity(A) = n, where A represents a map from F^n to F^m.

- If rank(A) = min(m,n), A has full rank; if A is square and full rank, it is invertible.

- Computation methods include reducing A to row echelon form, row/column operations, or using singular value decomposition (SVD).

Applications:

- Determining solvability of linear systems: Ax = b has a solution iff rank([A|b]) = rank(A).

- Describing the dimensions of solution spaces and image spaces.

- Useful in graph theory, numerical analysis, and data science for assessing independence and dimensionality.

Example:

Consider A = [ [1, 2, 3], [4, 5, 6] ]. A has rank 2, since its two rows are linearly independent, and min(m,n) = 2.

See also: rank-nullity theorem, row space, column space, singular value decomposition, matrix rank.