In mathematics, a totally positive matrix is a square matrix in which all the minors are positive: that is, the determinant of every square submatrix is a positive number.[1] A totally positive matrix has all entries positive, so it is also a positive matrix; and it has all principal minors positive (and positive eigenvalues). A symmetric totally positive matrix is therefore also positive-definite. A totally non-negative matrix is defined similarly, except that all the minors must be non-negative (positive or zero). Some authors use "totally positive" to include all totally non-negative matrices.
Let A = ( A i j ) i j {\displaystyle \mathbf {A} =(A_{ij})_{ij}} be an n × n matrix. Consider any p ∈ ∈ --> { 1 , 2 , … … --> , n } {\displaystyle p\in \{1,2,\ldots ,n\}} and any p × p submatrix of the form B = ( A i k j ℓ ℓ --> ) k ℓ ℓ --> {\displaystyle \mathbf {B} =(A_{i_{k}j_{\ell }})_{k\ell }} where:
Then A is a totally positive matrix if:[2]
for all submatrices B {\displaystyle \mathbf {B} } that can be formed this way.
Topics which historically led to the development of the theory of total positivity include the study of:[2]
For example, a Vandermonde matrix whose nodes are positive and increasing is a totally positive matrix.
This linear algebra-related article is a stub. You can help Wikipedia by expanding it.
Lokasi Pengunjung: 3.22.217.202