Spark (mathematics)
Encyclopedia
In mathematics
, specifically in linear algebra
, the spark of a matrix A is the smallest number n such that there exists a set of n columns in A which are linearly dependent. Formally,
By contrast, the rank
of a matrix is the smallest number k such that all sets of k + 1 columns in A are linearly dependent.
The concept of the spark is of use in the theory of compressive sensing, where requirements on the spark of the measurement matrix are used to ensure stability and consistency of various estimation techniques.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, specifically in linear algebra
Linear algebra
Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...
, the spark of a matrix A is the smallest number n such that there exists a set of n columns in A which are linearly dependent. Formally,
By contrast, the rank
Rank (linear algebra)
The column rank of a matrix A is the maximum number of linearly independent column vectors of A. The row rank of a matrix A is the maximum number of linearly independent row vectors of A...
of a matrix is the smallest number k such that all sets of k + 1 columns in A are linearly dependent.
The concept of the spark is of use in the theory of compressive sensing, where requirements on the spark of the measurement matrix are used to ensure stability and consistency of various estimation techniques.