Spark (mathematics)

In mathematics, specifically in linear algebra, the spark of a m × n {displaystyle m imes n} matrix A {displaystyle A} is the smallest number k {displaystyle k} such that there exists a set of k {displaystyle k} columns in A {displaystyle A} which are linearly dependent. Formally, s p a r k ( A ) = min d ≠ 0 ‖ d ‖ 0  s.t.  A d = 0 {displaystyle mathrm {spark} (A)=min _{d eq 0}|d|_{0}{ ext{ s.t. }}Ad=0}     (Eq.1) In mathematics, specifically in linear algebra, the spark of a m × n {displaystyle m imes n} matrix A {displaystyle A} is the smallest number k {displaystyle k} such that there exists a set of k {displaystyle k} columns in A {displaystyle A} which are linearly dependent. Formally, where d {displaystyle d} is a nonzero vector and ‖ d ‖ 0 {displaystyle |d|_{0}} denotes its number of nonzero coefficients. If all the columns are linearly independent, s p a r k ( A ) {displaystyle mathrm {spark} (A)} is usually defined to be ∞ {displaystyle infty } . By contrast, the rank of a matrix is the largest number k {displaystyle k} such that some set of k {displaystyle k} columns of A {displaystyle A} is linearly independent. Consider the following matrix A {displaystyle A} . A = [ 1 2 0 1 1 2 0 2 1 2 0 3 1 0 − 3 4 ] {displaystyle A={egin{bmatrix}1&2&0&1\1&2&0&2\1&2&0&3\1&0&-3&4end{bmatrix}}}