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). 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 be an n × n matrix. Consider any and any p × p submatrix of the form where:
Then A is a totally positive matrix if:
for all submatrices 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.