# Positive principal minors in non-symmetric matrix

Can anything be said about a (not necessarily symmetric) matrix $A$, all of whose principal minors (upper-left squares) have positive determinant? Do these matrices have a name?

I would like to know if this implies another condition usually found in positive-definite matrices, such as $x^T A x > 0$ for all $x \ne 0$, or that all eigenvalues are positive / have positive real part, etc.

• I am not entirely sure if this what you mean, but I think it might be what you're looking for: en.m.wikipedia.org/wiki/Sylvester's_criterion – dreamer Aug 9 '14 at 7:19
• @dreamer Sylvester's criterion deal with symmetric (or Hermitian) matrices. – Martin Sleziak Aug 9 '14 at 11:36
• A square matrix with every principal minor > 0 is called a P-matrix. The real eigen values of such a matrix are positive. – La Rias Mar 2 '16 at 9:50

This condition does not imply $x^TAx>0$ for all $x\ne 0$.
Define $A$ by $$A = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}.$$ Then all principal minors of $A$ have determinant $1$. But for $x\ne 0$ $$x^T Ax = \frac12 x^T(A+A^T)x = x^T \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}x.$$ For $x=\begin{pmatrix}1\\-1\end{pmatrix}$ this expression is zero.
• The O.P has asked whether one can any such property. I think the O.P is well aware of the fact that $x^TAx>0$ is not true. – Babai Aug 8 '16 at 22:09
• @Babai Please read again the question, in particular the part I would like to know if this implies another condition usually found in positive-definite matrices, such as xTAx>0 for all x≠0''. – daw Aug 9 '16 at 6:08
• If you read it fully the sentence doesn't stop with $x^TAx>0$ for all $x\neq 0$, it says more "......or eigen values are positive or positive real part." – Babai Aug 9 '16 at 6:23