What about the case $n=2?$ We can represent the family $\mathcal{F}$ by a graph with $m$ nodes labelled $v_1,\cdots,v_m$ (or unlabeled) and an edge for each putative basis. Then before including the information from column $1$ (aka $M_I$) we can say that the graph must be a complete multipartite graph possibly with some isolated vertices (call two non-zero rows equivalent if each is a multiple of the other, then the edges connect precisely the non-equivalent pairs.) Now the additional information from $M_I$ is which entries in column one are zero and which are not. The isolated vertices (if any) must come from the rows with a leading zero and the remaining rows (if any) with a leading zero form an equivalence class for the graph. Other than that the remaining classes can be anything.
About the case $n \gt 2$ I will just sayNote too that the information aboutfor $M_I$ tells us not only$j=1$ and $n=2$ we can do column operations on $M$ without changing which minors of $j \times j$ sub-matrices are nonsingular$M$ vanish. Each all $0$ row , but also the rank ofif any set of rows from, will always leave a $M_I.$ So if$0$ in column $1$ but otherwise we choosecan arrange to have no other $n$$0$ entries or to have the $0$'s indicate the all $0$ rows along with rankany other one equivalence class.
For $j$$j \gt 1$ I will just say that any (the maximum possible rank) then the corresponding$j$ rows of $M$ might or might not be a basis, however it definitely won't be a basis if the rank iswith span of dimension less than $j$ give a vanishing minor in $M_{I'}.$ I think one could show that by column operations in $M$ (which have no effect on which minors of $M$ vanish) we could arrange to have no other minors of of $M_I$ vanish.
What about the case $n=2?$ We can represent the family $\mathcal{F}$ by a graph with $m$ nodes labelled $v_1,\cdots,v_m$ (or unlabeled) and an edge for each putative basis. Then before including the information from column $1$ (aka $M_I$) we can say that the graph must be a complete multipartite graph possibly with some isolated vertices (call two non-zero rows equivalent if each is a multiple of the other, then the edges connect precisely the non-equivalent pairs.) Now the additional information from $M_I$ is which entries in column one are zero and which are not. The isolated vertices (if any) must come from the rows with a leading zero and the remaining rows (if any) with a leading zero form an equivalence class for the graph. Other than that the remaining classes can be anything.
About the case $n \gt 2$ I will just say that the information about $M_I$ tells us not only which $j \times j$ sub-matrices are nonsingular, but also the rank of any set of rows from $M_I.$ So if we choose $n$ rows with rank $j$ (the maximum possible rank) then the corresponding rows of $M$ might or might not be a basis, however it definitely won't be a basis if the rank is less than $j$.
What about the case $n=2?$ We can represent the family $\mathcal{F}$ by a graph with $m$ nodes labelled $v_1,\cdots,v_m$ (or unlabeled) and an edge for each putative basis. Then before including the information from column $1$ (aka $M_I$) we can say that the graph must be a complete multipartite graph possibly with some isolated vertices (call two non-zero rows equivalent if each is a multiple of the other, then the edges connect precisely the non-equivalent pairs.) Now the additional information from $M_I$ is which entries in column one are zero and which are not. The isolated vertices (if any) must come from the rows with a leading zero and the remaining rows (if any) with a leading zero form an equivalence class for the graph. Other than that the remaining classes can be anything.
Note too that for $j=1$ and $n=2$ we can do column operations on $M$ without changing which minors of $M$ vanish. Each all $0$ row , if any, will always leave a $0$ in column $1$ but otherwise we can arrange to have no other $0$ entries or to have the $0$'s indicate the all $0$ rows along with any other one equivalence class.
For $j \gt 1$ I will just say that any $j$ rows of $M$ with span of dimension less than $j$ give a vanishing minor in $M_{I'}.$ I think one could show that by column operations in $M$ (which have no effect on which minors of $M$ vanish) we could arrange to have no other minors of of $M_I$ vanish.
This seems like a matroidmatroid question but I'll take an elementary approach. It may be that my remarks just knock off the easy observations. I don't think anything changes if you switch to $\mathbb{Z}$ although fields of finite characteristic are different. We may as well assume that $M_{I'}=M$ and that $M_I$ is the $j$ leftmost columns.
BeforeSo we even factor in the information gleaned fromhave a set $M_I,$ there are restrictions on which maximal minors$S=\{{v_1,v_2,\cdots,v_m\}}$ of $M_{I'}$ can vanish$m$ symbols and a family (and the same restriction apply to the information we can get about$\mathcal{F}$ of $M_I$)$n$-subsets from $S$. If We wonder when there is a certain maximal minorway to assign the $v_i$ to be $m$ row vectors from $V=\mathbb{C}^n$ so that $\mathcal{F}$ is non-zero then there are these possibilitiesexactly the bases for each remaining row : Either it belongs to a non-vanishing minor with some$V$ made from $j-1$$S.$ That is an interesting question in and of itself. But you to think of the previous$v_i$ as rows of a matrix -Or- it is$M$ (so the zero row and any minor it participates in vanishes. So your question is perhapsequivalent to which minors vanish and which don't) and wonder what elsemore can we say given the additional information from $M_I?$ .
Extremes canWhatever is true for $M_I$, it could be goodthat there are no bases from rows of $M$ (i.e. all minors vanish). If so, that is the end of it. Otherwise we can call a subset of $S$ independent if it is a subset of a basis. Then it is required that given an independent set $T$ of size $k$ and a basis $B$, there is a basis made up of $T$ and some $n-k$ elements from $B$. So that is a non-trivial restriction and it applies to thinkthe information that we are provided about $M_I.$ However there are families $\mathcal{F}$ which meet this requirement and can be realized by matrices over $\mathbb{Z}_2$ but not over $\mathbb{C}$ and also families which can not be realized by a matrix over any field.
Assume $I$$M_I$ is the first $j$ columns. It will follow from below that
(*) if every minor for $M_I$ vanishes, the same is true for $M_{I'}$$M$ and that
(* *) otherwise we may assume that the top minor of $M_I$ is the $j \times j$ identity matrix and
(* * *) that we know which entries of $M_I$ are zero and that the leading non-zero entry of each row is a $1.$
I don't actually make use of these last two but they are helpful to think about.
- Whatever is true for $M_{I}$ it could happen that every minor of $M_{I'}$ vanishes (if all the other columns are in the span of $I$).
- If $j=1$ then we know which entries in column $1$ are zero. A minor of $M_{I'}$ with all zero in the first column will vanish. Any other first column might or might not lead to a non-zero minor.
If $j=1$ then we know which entries in column $1$ are zero. A minor of $M$ with all zero in the first column will vanish. Any other first column might or might not lead to a non-zero minor.
- In general, a minor for $M_{I'}$ with all leftmost $j \times j$ minors vanishing also vanishes (hence (*)).
In general, a minor for $M$ with all leftmost $j \times j$ minors vanishing also vanishes (hence (*)). What else can one say? I argue below that the answer is nothing more for the case $n=2.$ Otherwise I can think of a bit more, but not much.
Column operations on $M_{I'}$ will affect the values of the minors but not which ones vanish. And permuting the rows is just a relabelling which does not change the problem. Hence (* * ). And then just the minors which use $j-1$ of the top $j$ rows will tell you the rest of (* * *).
I suppose another extreme is $m=n+1$. We already saw that it might be that every minor of $M_{I'}$ vanishes and that otherwise we may assume that the top $j$ rows of $M_{I'}$ agree with the $(j+1) \times (j+1)$ identity matrix. We then know which entries of $M_{I'}$ are zero and which are not simply from (* * *) above and since multiplying a row by a (in the final column) from the vanishing or nonnon-vanishing of maximal $M_{I'}$ minors using the top $j$ rowszero) constant does not change anything.
What about the case $n=2?$ We can represent the family $\mathcal{F}$ by a graph with $m$ nodes labelled $v_1,\cdots,v_m$ (or unlabeled) and an edge for each putative basis. Then before including the information from column $1$ (aka $M_I$) we can say that the graph must be a complete multipartite graph possibly with some isolated vertices (call two non-zero rows equivalent if each is a multiple of the other, then the edges connect precisely the non-equivalent pairs.) Now the additional information from $M_I$ is which entries in column one are zero and which are not. The isolated vertices (if any) must come from the rows with a leading zero and the remaining rows (if any) with a leading zero form an equivalence class for the graph. Other than that the remaining classes can be anything.
About the case $n \gt 2$ I will just say that the information about $M_I$ tells us not only which $j \times j$ sub-matrices are nonsingular, but also the rank of any set of rows from $M_I.$ So if we choose $n$ rows with rank $j$ (the maximum possible rank) then the corresponding rows of $M$ might or might not be a basis, however it definitely won't be a basis if the rank is less than $j$.
This seems like a matroid question but I'll take an elementary approach. It may be that my remarks just knock off the easy observations.
Before we even factor in the information gleaned from $M_I,$ there are restrictions on which maximal minors of $M_{I'}$ can vanish (and the same restriction apply to the information we can get about $M_I$). If a certain maximal minor is non-zero then there are these possibilities for each remaining row : Either it belongs to a non-vanishing minor with some $j-1$ of the previous rows -Or- it is the zero row and any minor it participates in vanishes. So your question is perhaps what else can we say given the additional information from $M_I?$ .
Extremes can be good to think about. Assume $I$ is the first $j$ columns. It will follow from below that
(*) if every minor for $M_I$ vanishes, the same is true for $M_{I'}$ and that
(* *) otherwise we may assume that the top minor is the $j \times j$ identity matrix and
(* * *) that we know which entries of $M_I$ are zero.
- Whatever is true for $M_{I}$ it could happen that every minor of $M_{I'}$ vanishes (if all the other columns are in the span of $I$).
- If $j=1$ then we know which entries in column $1$ are zero. A minor of $M_{I'}$ with all zero in the first column will vanish. Any other first column might or might not lead to a non-zero minor.
- In general, a minor for $M_{I'}$ with all leftmost $j \times j$ minors vanishing also vanishes (hence (*)).
Column operations on $M_{I'}$ will affect the values of the minors but not which ones vanish. And permuting the rows is just a relabelling which does not change the problem. Hence (* * ). And then just the minors which use $j-1$ of the top $j$ rows will tell you the rest of (* * *).
I suppose another extreme is $m=n+1$. We already saw that it might be that every minor of $M_{I'}$ vanishes and that otherwise we may assume that the top $j$ rows of $M_{I'}$ agree with the $(j+1) \times (j+1)$ identity matrix. We then know which entries of $M_{I'}$ are zero and which are not simply from (* * *) above and (in the final column) from the vanishing or non-vanishing of maximal $M_{I'}$ minors using the top $j$ rows.
This seems like a matroid question but I'll take an elementary approach. It may be that my remarks just knock off the easy observations. I don't think anything changes if you switch to $\mathbb{Z}$ although fields of finite characteristic are different. We may as well assume that $M_{I'}=M$ and that $M_I$ is the $j$ leftmost columns.
So we have a set $S=\{{v_1,v_2,\cdots,v_m\}}$ of $m$ symbols and a family $\mathcal{F}$ of $n$-subsets from $S$. We wonder when there is a way to assign the $v_i$ to be $m$ row vectors from $V=\mathbb{C}^n$ so that $\mathcal{F}$ is exactly the bases for $V$ made from $S.$ That is an interesting question in and of itself. But you to think of the $v_i$ as rows of a matrix $M$ (so the question is equivalent to which minors vanish and which don't) and wonder what more can we say given the additional information from $M_I?$
Whatever is true for $M_I$, it could be that there are no bases from rows of $M$ (i.e. all minors vanish). If so, that is the end of it. Otherwise we can call a subset of $S$ independent if it is a subset of a basis. Then it is required that given an independent set $T$ of size $k$ and a basis $B$, there is a basis made up of $T$ and some $n-k$ elements from $B$. So that is a non-trivial restriction and it applies to the information that we are provided about $M_I.$ However there are families $\mathcal{F}$ which meet this requirement and can be realized by matrices over $\mathbb{Z}_2$ but not over $\mathbb{C}$ and also families which can not be realized by a matrix over any field.
Assume $M_I$ is the first $j$ columns. It will follow from below that
(*) if every minor for $M_I$ vanishes, the same is true for $M$ and that
(* *) otherwise we may assume that the top minor of $M_I$ is the $j \times j$ identity matrix and
(* * *) that we know which entries of $M_I$ are zero and that the leading non-zero entry of each row is a $1.$
I don't actually make use of these last two but they are helpful to think about.
If $j=1$ then we know which entries in column $1$ are zero. A minor of $M$ with all zero in the first column will vanish. Any other first column might or might not lead to a non-zero minor.
In general, a minor for $M$ with all leftmost $j \times j$ minors vanishing also vanishes (hence (*)). What else can one say? I argue below that the answer is nothing more for the case $n=2.$ Otherwise I can think of a bit more, but not much.
Column operations on $M_{I'}$ will affect the values of the minors but not which ones vanish. And permuting the rows is just a relabelling which does not change the problem. Hence (* * ). And then just the minors which use $j-1$ of the top $j$ rows will tell you the rest of (* * *) since multiplying a row by a (non-zero) constant does not change anything.
What about the case $n=2?$ We can represent the family $\mathcal{F}$ by a graph with $m$ nodes labelled $v_1,\cdots,v_m$ (or unlabeled) and an edge for each putative basis. Then before including the information from column $1$ (aka $M_I$) we can say that the graph must be a complete multipartite graph possibly with some isolated vertices (call two non-zero rows equivalent if each is a multiple of the other, then the edges connect precisely the non-equivalent pairs.) Now the additional information from $M_I$ is which entries in column one are zero and which are not. The isolated vertices (if any) must come from the rows with a leading zero and the remaining rows (if any) with a leading zero form an equivalence class for the graph. Other than that the remaining classes can be anything.
About the case $n \gt 2$ I will just say that the information about $M_I$ tells us not only which $j \times j$ sub-matrices are nonsingular, but also the rank of any set of rows from $M_I.$ So if we choose $n$ rows with rank $j$ (the maximum possible rank) then the corresponding rows of $M$ might or might not be a basis, however it definitely won't be a basis if the rank is less than $j$.
This seems like a matroid question but I'll take an elementary approach. It may be that my remarks just knock off the easy observations.
Before we even factor in the information gleaned from $M_I,$ there are restrictions on which maximal minors of $M_{I'}$ can vanish (and the same restriction apply to the information we can get about $M_I$). If a certain maximal minor is non-zero then there are these possibilities for each remaining row : Either it belongs to a non-vanishing minor with some $j-1$ of the previous rows -Or- it is the zero row and any minor it participates in vanishes. So your question is perhaps what else can we say given the additional information from $M_I?$ .
Extremes can be good to think about. Assume $I$ is the first $j$ columns. It will follow from below that
(*) if every minor for $M_I$ vanishes, the same is true for $M_{I'}$ and that
(* *) otherwise we may assume that the top minor is the $j \times j$ identity matrix and
(* * *) that we know which entries of $M_I$ are zero.
- Whatever is true for $M_{I}$ it could happen that every minor of $M_{I'}$ vanishes (if all the other columns are in the span of $I$).
- If $j=1$ then we know which entries in column $1$ are zero. A minor of $M_{I'}$ with all zero in the first column will vanish. Any other first column might or might not lead to a non-zero minor.
- In general, a minor for $M_{I'}$ with all leftmost $j \times j$ minors vanishing also vanishes (hence (*)).
Column operations on $M_{I'}$ will affect the values of the minors but not which ones vanish. And permuting the rows is just a relabelling which does not change the problem. Hence (* * ). And then just the minors which use $j-1$ of the top $j$ rows will tell you the rest of (* * *).
There is the question of what the vanishing pattern can be for $M_I$.
So your question is perhaps what else can we say given the additional information from $M_I?$
I suppose another extreme is $m=n+1$. We already saw that it might be that every minor of $M_{I'}$ vanishes and that otherwise we may assume that the top $j$ rows of $M_{I'}$ agree with the $(j+1) \times (j+1)$ identity matrix. We then know which entries of $M_{I'}$ are zero and which are not simply from (* * *) above and (in the final column) from the vanishing or non-vanishing of maximal $M_{I'}$ minors using the top $j$ rows.
This seems like a matroid question but I'll take an elementary approach. It may be that my remarks just knock off the easy observations.
Before we even factor in the information gleaned from $M_I,$ there are restrictions on which maximal minors of $M_{I'}$ can vanish (and the same restriction apply to the information we can get about $M_I$). If a certain maximal minor is non-zero then there are these possibilities for each remaining row : Either it belongs to a non-vanishing minor with some $j-1$ of the previous rows -Or- it is the zero row and any minor it participates in vanishes. So your question is perhaps what else can we say given the additional information from $M_I?$ .
Extremes can be good to think about. Assume $I$ is the first $j$ columns. It will follow from below that
(*) if every minor for $M_I$ vanishes, the same is true for $M_{I'}$ and that
(* *) otherwise we may assume that the top minor is the $j \times j$ identity matrix and
(* * *) that we know which entries of $M_I$ are zero.
- Whatever is true for $M_{I}$ it could happen that every minor of $M_{I'}$ vanishes (if all the other columns are in the span of $I$).
- If $j=1$ then we know which entries in column $1$ are zero. A minor of $M_{I'}$ with all zero in the first column will vanish. Any other first column might or might not lead to a non-zero minor.
- In general, a minor for $M_{I'}$ with all leftmost $j \times j$ minors vanishing also vanishes (hence (*)).
Column operations on $M_{I'}$ will affect the values of the minors but not which ones vanish. And permuting the rows is just a relabelling which does not change the problem. Hence (* * ). And then just the minors which use $j-1$ of the top $j$ rows will tell you the rest of (* * *).
There is the question of what the vanishing pattern can be for $M_I$.
So your question is perhaps what else can we say given the additional information from $M_I?$
I suppose another extreme is $m=n+1$. We already saw that it might be that every minor of $M_{I'}$ vanishes and that otherwise we may assume that the top $j$ rows of $M_{I'}$ agree with the $(j+1) \times (j+1)$ identity matrix. We then know which entries of $M_{I'}$ are zero and which are not simply from (* * *) above and (in the final column) from the vanishing or non-vanishing of maximal $M_{I'}$ minors using the top $j$ rows.
This seems like a matroid question but I'll take an elementary approach. It may be that my remarks just knock off the easy observations.
Before we even factor in the information gleaned from $M_I,$ there are restrictions on which maximal minors of $M_{I'}$ can vanish (and the same restriction apply to the information we can get about $M_I$). If a certain maximal minor is non-zero then there are these possibilities for each remaining row : Either it belongs to a non-vanishing minor with some $j-1$ of the previous rows -Or- it is the zero row and any minor it participates in vanishes. So your question is perhaps what else can we say given the additional information from $M_I?$ .
Extremes can be good to think about. Assume $I$ is the first $j$ columns. It will follow from below that
(*) if every minor for $M_I$ vanishes, the same is true for $M_{I'}$ and that
(* *) otherwise we may assume that the top minor is the $j \times j$ identity matrix and
(* * *) that we know which entries of $M_I$ are zero.
- Whatever is true for $M_{I}$ it could happen that every minor of $M_{I'}$ vanishes (if all the other columns are in the span of $I$).
- If $j=1$ then we know which entries in column $1$ are zero. A minor of $M_{I'}$ with all zero in the first column will vanish. Any other first column might or might not lead to a non-zero minor.
- In general, a minor for $M_{I'}$ with all leftmost $j \times j$ minors vanishing also vanishes (hence (*)).
Column operations on $M_{I'}$ will affect the values of the minors but not which ones vanish. And permuting the rows is just a relabelling which does not change the problem. Hence (* * ). And then just the minors which use $j-1$ of the top $j$ rows will tell you the rest of (* * *).
I suppose another extreme is $m=n+1$. We already saw that it might be that every minor of $M_{I'}$ vanishes and that otherwise we may assume that the top $j$ rows of $M_{I'}$ agree with the $(j+1) \times (j+1)$ identity matrix. We then know which entries of $M_{I'}$ are zero and which are not simply from (* * *) above and (in the final column) from the vanishing or non-vanishing of maximal $M_{I'}$ minors using the top $j$ rows.