Questions tagged [m-matrix]

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9
votes
2answers
558 views

Product $PVPVP$ is elementwise nonnegative?

Let $P\in \mathbb{R}^{n\times n}$ be the inverse of a positive definite M-matrix and $V\in \mathbb{R}^{n\times n}$ be any diagonal matrix. Prove (or disprove) that $PVPVP$ is elementwise nonnegative. ...
6
votes
1answer
440 views

Is there an efficient algorithm to check whether a matrix is symmetrizable using only permutation matrix?

Is there any known efficient algorithm (something that works better than brute-force algorithm) to check that given a $(0,1)$-$d \times d$ matrix $A$, is there a permutation matrix $P$ of the same ...
4
votes
2answers
108 views

Requesting reference for result from linear algebra on Schur complements of M-matrices

In my research in linear algebra, I have come across a useful result stating that the Schur complement of a principal non-singular submatrix of an M-matrix is also an M-matrix, but I have never found ...
3
votes
0answers
134 views

inverse M-matrix times mixed-sign vector

Recently a colleague and I came across this unusual phenomenon. Take $M\in\mathbb{R}^{n\times n}$ a singular irreducible M-matrix, and $b\in\mathbb{R}^{n}$ such that the system $Mx=b$ is solvable (so,...
2
votes
1answer
573 views

When is a Schur complement an $M$-matrix?

Let $F=\begin{bmatrix}A & B \\\\ B^{T} & D\end{bmatrix}$ be symmetric and strictly diagonally dominant (thus an $H$-matrix). I also know that $B>0$ entrywise. What I am trying to show is ...
2
votes
1answer
1k views

Irreducible non-singular M-matrices and complex numbers

It is well known that a non-singular M-matrix that is irreducible has a strictly positive inverse (all entries $>0$). An M-matrix is a matrix that has eigenvalues with positive real part, and the ...
1
vote
1answer
142 views

Find a matrix and its inverse satisfying lower and upper bounds

I reduced a problem of matrix completion to the problem find $A,B$ such that $AB=I$ $A_{min}\leq A \leq A_{max}$ $B_{min}\leq B \leq B_{max}$ One possible approach would be to just minimize $\|AB-...
1
vote
0answers
263 views

M-matrix with nonconstant entries properties

I have a matrix $J(x)$ with $J_{ij}(x)=f_{ij}(x)$ where vector $x$ is $x=x_1, x_2, ..., x_m$. I have shown that $J(x)$ is an M-matrix for all $x$. There is known review paper by Plemmons (1977) of 40 ...