Linear and multilinear algebra; matrix theory Definition

Totally positive matrix

Definition

Let $A$ be an $m \times n$ matrix with real entries. The matrix $A$ is totally positive when every minor of $A$ is positive. That is, for all $1 \le k \le \min(m, n)$ , all rows $i_1 < \cdots < i_k$ and all columns $j_1 < \cdots < j_k$ , the $k \times k$ submatrix on these rows and columns has positive determinant.

Remarks

Example: $\begin{pmatrix} 1 & 1 \\ 1 & 2 \end{pmatrix}$ is totally positive: its $1 \times 1$ minors $1, 1, 1, 2$ are positive, and its only $2 \times 2$ minor is $2 - 1 = 1 > 0$ . Non-example: $\begin{pmatrix} 1 & 2 \\ 2 & 1 \end{pmatrix}$ is not totally positive: all entries are positive, but the determinant is $1 - 4 = -3$ . The matrix is totally nonnegative when every minor is nonnegative. Some authors use "totally positive" for this weaker condition and "strictly totally positive" for the condition above. The Vandermonde matrix of real numbers $0 < x_1 < \cdots < x_n$ is totally positive.

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Matrix concept
Minor concept
Totally positive matrix definition

Tag	`BTRNZKN1BP`
URI	`m2://BTRNZKN1BP`
Kind	definition
Writing	W0 stub
MSC	15B48
Defines	Totally positive matrix
Uses	Matrix, Minor
Changed	2 November 2026

Totally positive matrix

Definition

Remarks

Reading path

Further reading

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