Totally positive matrix

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. 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), so it is also a positive-definite matrix. 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. 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. 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), so it is also a positive-definite matrix. 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 pin {1,2,ldots ,n}} and any p × p submatrix of the form B = ( A i k j ℓ ) k ℓ {displaystyle mathbf {B} =(A_{i_{k}j_{ell }})_{kell }} where: Then A is a totally positive matrix if: 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: For example, a Vandermonde matrix whose nodes are positive and increasing is a totally positive matrix.