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
be an *n* × *n* matrix. Consider any and any *p* × *p* submatrix of the form
where:

Then **A** is a **totally positive matrix** if:^{[2]}

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:^{[2]}

- the spectral properties of kernels and matrices which are totally positive,
- ordinary differential equations whose Green's function is totally positive (by M. G. Krein and some colleagues in the mid-1930s),
- the variation diminishing properties (started by I. J. Schoenberg in 1930),
- Pólya frequency functions (by I. J. Schoenberg in the late 1940s and early 1950s).

For example, a Vandermonde matrix whose nodes are positive and increasing is a totally positive matrix.