Dimension of biggest linear subspace inside a variety of matrices
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A variety of matrices (or algebraic set) $V$ is a set of matrices over a field $F$ which are the common zeros of a set of polynomials (on their entries). Some varieties contain linear subspaces of matrices, and others do not. Examples:
- Every linear subspace is a variety.
- If $0notin V$ then there is no linear subspace inside $V$.
- If $det(A)=0$ then $det(lambda A)=0$ for all $lambdain F$.
- If $V$ is the variety of $det$ on $textMat_n(F)$ and $L$ denotes the subspace of all matrices whose first row is full of zeros, then $Lsubseteq V$ and $dim_F L=n^2-n$.
Given a variety $V$ inside $textMat_mtimes n(F)$, define
$$textldim(V):=max LleqtextMat_mtimes n(F), Lsubseteq V$$
and ldim$(V)=-1$ if there is no such $L$ (iff $0notin V$).
My questions are the following:
1) Given $V$, what tools do we have to find ldim$(V)$?
2) What do we know when $V$ are the determinantal varieties (those of matrices with rank no greater than a fixed $k$)?
3) Given $textMat_mtimes n(F)$, what is the maximum ldim possible for varieties which are not subspaces themselves, and for which varieties is it attained? For square matrices, is it $n^2-n$ as in the third example?
linear-algebra matrices algebraic-geometry
asked Aug 14 at 10:30
Jose Brox
1,9051820
add a comment |Â
up vote
1
down vote
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A variety of matrices (or algebraic set) $V$ is a set of matrices over a field $F$ which are the common zeros of a set of polynomials (on their entries). Some varieties contain linear subspaces of matrices, and others do not. Examples:
- Every linear subspace is a variety.
- If $0notin V$ then there is no linear subspace inside $V$.
- If $det(A)=0$ then $det(lambda A)=0$ for all $lambdain F$.
- If $V$ is the variety of $det$ on $textMat_n(F)$ and $L$ denotes the subspace of all matrices whose first row is full of zeros, then $Lsubseteq V$ and $dim_F L=n^2-n$.
Given a variety $V$ inside $textMat_mtimes n(F)$, define
$$textldim(V):=max LleqtextMat_mtimes n(F), Lsubseteq V$$
and ldim$(V)=-1$ if there is no such $L$ (iff $0notin V$).
My questions are the following:
1) Given $V$, what tools do we have to find ldim$(V)$?
2) What do we know when $V$ are the determinantal varieties (those of matrices with rank no greater than a fixed $k$)?
3) Given $textMat_mtimes n(F)$, what is the maximum ldim possible for varieties which are not subspaces themselves, and for which varieties is it attained? For square matrices, is it $n^2-n$ as in the third example?
linear-algebra matrices algebraic-geometry
asked Aug 14 at 10:30
Jose Brox
1,9051820
I don't know the answer to your questions, but you might find this paper useful: mathscinet.ams.org/mathscinet-getitem?mr=1689267
– Asal Beag Dubh
Aug 14 at 11:27
Nice question. But does it matter that we think about matrices? Maybe this is distracting. Your definitions work as well for any algebraic set $A subset F^n$.
– red_trumpet
Aug 14 at 20:55
1
@red_trumpet Indeed, you are right, it could be that the answers are easier in the more general setting (perhaps someone knows some general theorem about this?). But I hope that, given the additional structure of matrix rings (rank, matrix groups, symmetric functions, etc.) this may have already been developed in the literature; I know of similar results (like if a subspace has an element of rank greater than k, then it must have dimension greater than nk). Another reason is that these questions have some relevance to linear preserver problems, a subject in which I'm interested in.
– Jose Brox
Aug 15 at 6:46
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
A variety of matrices (or algebraic set) $V$ is a set of matrices over a field $F$ which are the common zeros of a set of polynomials (on their entries). Some varieties contain linear subspaces of matrices, and others do not. Examples:
- Every linear subspace is a variety.
- If $0notin V$ then there is no linear subspace inside $V$.
- If $det(A)=0$ then $det(lambda A)=0$ for all $lambdain F$.
- If $V$ is the variety of $det$ on $textMat_n(F)$ and $L$ denotes the subspace of all matrices whose first row is full of zeros, then $Lsubseteq V$ and $dim_F L=n^2-n$.
Given a variety $V$ inside $textMat_mtimes n(F)$, define
$$textldim(V):=max LleqtextMat_mtimes n(F), Lsubseteq V$$
and ldim$(V)=-1$ if there is no such $L$ (iff $0notin V$).
My questions are the following:
1) Given $V$, what tools do we have to find ldim$(V)$?
2) What do we know when $V$ are the determinantal varieties (those of matrices with rank no greater than a fixed $k$)?
3) Given $textMat_mtimes n(F)$, what is the maximum ldim possible for varieties which are not subspaces themselves, and for which varieties is it attained? For square matrices, is it $n^2-n$ as in the third example?
linear-algebra matrices algebraic-geometry
asked Aug 14 at 10:30
Jose Brox
1,9051820
A variety of matrices (or algebraic set) $V$ is a set of matrices over a field $F$ which are the common zeros of a set of polynomials (on their entries). Some varieties contain linear subspaces of matrices, and others do not. Examples:
- Every linear subspace is a variety.
- If $0notin V$ then there is no linear subspace inside $V$.
- If $det(A)=0$ then $det(lambda A)=0$ for all $lambdain F$.
- If $V$ is the variety of $det$ on $textMat_n(F)$ and $L$ denotes the subspace of all matrices whose first row is full of zeros, then $Lsubseteq V$ and $dim_F L=n^2-n$.
Given a variety $V$ inside $textMat_mtimes n(F)$, define
$$textldim(V):=max LleqtextMat_mtimes n(F), Lsubseteq V$$
and ldim$(V)=-1$ if there is no such $L$ (iff $0notin V$).
My questions are the following:
1) Given $V$, what tools do we have to find ldim$(V)$?
2) What do we know when $V$ are the determinantal varieties (those of matrices with rank no greater than a fixed $k$)?
3) Given $textMat_mtimes n(F)$, what is the maximum ldim possible for varieties which are not subspaces themselves, and for which varieties is it attained? For square matrices, is it $n^2-n$ as in the third example?
linear-algebra matrices algebraic-geometry
asked Aug 14 at 10:30
Jose Brox
1,9051820
edited Aug 14 at 10:47
asked Aug 14 at 10:30
Jose Brox
1,9051820
asked Aug 14 at 10:30
Jose Brox
1,9051820
asked Aug 14 at 10:30
Jose Brox
1,9051820
1,9051820
I don't know the answer to your questions, but you might find this paper useful: mathscinet.ams.org/mathscinet-getitem?mr=1689267
– Asal Beag Dubh
Aug 14 at 11:27
Nice question. But does it matter that we think about matrices? Maybe this is distracting. Your definitions work as well for any algebraic set $A subset F^n$.
– red_trumpet
Aug 14 at 20:55
1
@red_trumpet Indeed, you are right, it could be that the answers are easier in the more general setting (perhaps someone knows some general theorem about this?). But I hope that, given the additional structure of matrix rings (rank, matrix groups, symmetric functions, etc.) this may have already been developed in the literature; I know of similar results (like if a subspace has an element of rank greater than k, then it must have dimension greater than nk). Another reason is that these questions have some relevance to linear preserver problems, a subject in which I'm interested in.
– Jose Brox
Aug 15 at 6:46
add a comment |Â
I don't know the answer to your questions, but you might find this paper useful: mathscinet.ams.org/mathscinet-getitem?mr=1689267
– Asal Beag Dubh
Aug 14 at 11:27
Nice question. But does it matter that we think about matrices? Maybe this is distracting. Your definitions work as well for any algebraic set $A subset F^n$.
– red_trumpet
Aug 14 at 20:55
1
@red_trumpet Indeed, you are right, it could be that the answers are easier in the more general setting (perhaps someone knows some general theorem about this?). But I hope that, given the additional structure of matrix rings (rank, matrix groups, symmetric functions, etc.) this may have already been developed in the literature; I know of similar results (like if a subspace has an element of rank greater than k, then it must have dimension greater than nk). Another reason is that these questions have some relevance to linear preserver problems, a subject in which I'm interested in.
– Jose Brox
Aug 15 at 6:46
I don't know the answer to your questions, but you might find this paper useful: mathscinet.ams.org/mathscinet-getitem?mr=1689267
– Asal Beag Dubh
Aug 14 at 11:27
I don't know the answer to your questions, but you might find this paper useful: mathscinet.ams.org/mathscinet-getitem?mr=1689267
– Asal Beag Dubh
Aug 14 at 11:27
Nice question. But does it matter that we think about matrices? Maybe this is distracting. Your definitions work as well for any algebraic set $A subset F^n$.
– red_trumpet
Aug 14 at 20:55
Nice question. But does it matter that we think about matrices? Maybe this is distracting. Your definitions work as well for any algebraic set $A subset F^n$.
– red_trumpet
Aug 14 at 20:55
1
1
@red_trumpet Indeed, you are right, it could be that the answers are easier in the more general setting (perhaps someone knows some general theorem about this?). But I hope that, given the additional structure of matrix rings (rank, matrix groups, symmetric functions, etc.) this may have already been developed in the literature; I know of similar results (like if a subspace has an element of rank greater than k, then it must have dimension greater than nk). Another reason is that these questions have some relevance to linear preserver problems, a subject in which I'm interested in.
– Jose Brox
Aug 15 at 6:46
@red_trumpet Indeed, you are right, it could be that the answers are easier in the more general setting (perhaps someone knows some general theorem about this?). But I hope that, given the additional structure of matrix rings (rank, matrix groups, symmetric functions, etc.) this may have already been developed in the literature; I know of similar results (like if a subspace has an element of rank greater than k, then it must have dimension greater than nk). Another reason is that these questions have some relevance to linear preserver problems, a subject in which I'm interested in.
– Jose Brox
Aug 15 at 6:46
add a comment |Â
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I don't know the answer to your questions, but you might find this paper useful: mathscinet.ams.org/mathscinet-getitem?mr=1689267
– Asal Beag Dubh
Aug 14 at 11:27
Nice question. But does it matter that we think about matrices? Maybe this is distracting. Your definitions work as well for any algebraic set $A subset F^n$.
– red_trumpet
Aug 14 at 20:55
1
@red_trumpet Indeed, you are right, it could be that the answers are easier in the more general setting (perhaps someone knows some general theorem about this?). But I hope that, given the additional structure of matrix rings (rank, matrix groups, symmetric functions, etc.) this may have already been developed in the literature; I know of similar results (like if a subspace has an element of rank greater than k, then it must have dimension greater than nk). Another reason is that these questions have some relevance to linear preserver problems, a subject in which I'm interested in.
– Jose Brox
Aug 15 at 6:46