P-Matrix -- from Wolfram MathWorld

P-Matrix

A real -matrix is a real matrix for which every nonempty principal submatrix has a positive determinant (Fallat and Tsatsomeros 2002, p. 190). The matrix need not be symmetric. For a symmetric matrix, this condition is equivalent to positive definiteness.

Every real -matrix has no eigenvalue in , so its principal matrix power is defined for every real number . Xie (2026, Theorem 2.1) determined the exponents preserving the class in each order. If is the set of real numbers such that is a -matrix for every real -matrix , then

$E_n={R for n=1; [-1,1] for n=2 or n=3; {-1,0,1} for n>=4.$

(1)

These are principal matrix powers, not entrywise powers. The restrictions concern all real -matrices of a fixed order, rather than one fixed matrix.

References

Fallat, S. M. and Tsatsomeros, M. J. "On the Cayley Transform of Positivity Classes of Matrices." Elec. J. Linear Algebra 9, 190-196, 2002. https://doi.org/10.13001/1081-3810.1086.

Xie, W. "Complete Bernstein Functions and Real Powers of

-Matrices." 25 Oct 2026. https://arxiv.org/abs/2611.00096v1.

P-Matrix

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