Linear combination of vectors with norming term to be a matrix

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For two real vectors $X_1$ and $X_2$, by "linear combination", we mean $aX_1 + bX_2$ for any $a$ and $b$. We use this term while defining vector spaces and related things.



Is there any standard name for such combination: $AX_1 + BX_2$, where $A$ and $B$ are real matrices? Like "Linear matrix combination"? Is there any standard literature where people have investigated the properties of such combination?




linear-algebra vector-spaces vectors linear-transformations vector-analysis




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  • This may be useful math.stackexchange.com/questions/636653/…
    – AnyAD
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up vote
2
down vote

favorite












For two real vectors $X_1$ and $X_2$, by "linear combination", we mean $aX_1 + bX_2$ for any $a$ and $b$. We use this term while defining vector spaces and related things.



Is there any standard name for such combination: $AX_1 + BX_2$, where $A$ and $B$ are real matrices? Like "Linear matrix combination"? Is there any standard literature where people have investigated the properties of such combination?




linear-algebra vector-spaces vectors linear-transformations vector-analysis




share|cite|improve this question



















  • This may be useful math.stackexchange.com/questions/636653/…
    – AnyAD
    Jul 16 at 0:07












up vote
2
down vote

favorite









up vote
2
down vote

favorite











For two real vectors $X_1$ and $X_2$, by "linear combination", we mean $aX_1 + bX_2$ for any $a$ and $b$. We use this term while defining vector spaces and related things.



Is there any standard name for such combination: $AX_1 + BX_2$, where $A$ and $B$ are real matrices? Like "Linear matrix combination"? Is there any standard literature where people have investigated the properties of such combination?




linear-algebra vector-spaces vectors linear-transformations vector-analysis




share|cite|improve this question











For two real vectors $X_1$ and $X_2$, by "linear combination", we mean $aX_1 + bX_2$ for any $a$ and $b$. We use this term while defining vector spaces and related things.



Is there any standard name for such combination: $AX_1 + BX_2$, where $A$ and $B$ are real matrices? Like "Linear matrix combination"? Is there any standard literature where people have investigated the properties of such combination?





linear-algebra vector-spaces vectors linear-transformations vector-analysis





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share|cite|improve this question




share|cite|improve this question









asked Jul 15 at 23:31









Joy

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  • This may be useful math.stackexchange.com/questions/636653/…
    – AnyAD
    Jul 16 at 0:07
















  • This may be useful math.stackexchange.com/questions/636653/…
    – AnyAD
    Jul 16 at 0:07















This may be useful math.stackexchange.com/questions/636653/…
– AnyAD
Jul 16 at 0:07




This may be useful math.stackexchange.com/questions/636653/…
– AnyAD
Jul 16 at 0:07










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'Evaluating' $AX_1$, for all $X_1$ will give the column space of $A $, and similarly for $B$. The sum of these two will give the set of vectors $u+v$, wher $u,v $ are elements of column space of $A $ and clmn space of $B $ respectively. That is, we get the (algebraic) sum of the two subspaces (also a subspace). Considering the matrices as operators this is also obviously $Im (A)+Im (B) $.



Sum of such subspaces and similar have been and are studied, and will be considered for example in operator theory. Books on Linear algebra may consider this and related questions.






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  • Thanks, @AnyAD. I am not a mathematician, I asked this question as such combination appeared in my research. So, one sub-question, are there any necessary conditions on $A$ and $B$ (like positive definiteness, non-singularity etc) in operator theory?
    – Joy
    Jul 16 at 8:37










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1 Answer
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1 Answer
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active

oldest

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active

oldest

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active

oldest

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up vote
0
down vote













'Evaluating' $AX_1$, for all $X_1$ will give the column space of $A $, and similarly for $B$. The sum of these two will give the set of vectors $u+v$, wher $u,v $ are elements of column space of $A $ and clmn space of $B $ respectively. That is, we get the (algebraic) sum of the two subspaces (also a subspace). Considering the matrices as operators this is also obviously $Im (A)+Im (B) $.



Sum of such subspaces and similar have been and are studied, and will be considered for example in operator theory. Books on Linear algebra may consider this and related questions.






share|cite|improve this answer























  • Thanks, @AnyAD. I am not a mathematician, I asked this question as such combination appeared in my research. So, one sub-question, are there any necessary conditions on $A$ and $B$ (like positive definiteness, non-singularity etc) in operator theory?
    – Joy
    Jul 16 at 8:37














up vote
0
down vote













'Evaluating' $AX_1$, for all $X_1$ will give the column space of $A $, and similarly for $B$. The sum of these two will give the set of vectors $u+v$, wher $u,v $ are elements of column space of $A $ and clmn space of $B $ respectively. That is, we get the (algebraic) sum of the two subspaces (also a subspace). Considering the matrices as operators this is also obviously $Im (A)+Im (B) $.



Sum of such subspaces and similar have been and are studied, and will be considered for example in operator theory. Books on Linear algebra may consider this and related questions.






share|cite|improve this answer























  • Thanks, @AnyAD. I am not a mathematician, I asked this question as such combination appeared in my research. So, one sub-question, are there any necessary conditions on $A$ and $B$ (like positive definiteness, non-singularity etc) in operator theory?
    – Joy
    Jul 16 at 8:37












up vote
0
down vote










up vote
0
down vote









'Evaluating' $AX_1$, for all $X_1$ will give the column space of $A $, and similarly for $B$. The sum of these two will give the set of vectors $u+v$, wher $u,v $ are elements of column space of $A $ and clmn space of $B $ respectively. That is, we get the (algebraic) sum of the two subspaces (also a subspace). Considering the matrices as operators this is also obviously $Im (A)+Im (B) $.



Sum of such subspaces and similar have been and are studied, and will be considered for example in operator theory. Books on Linear algebra may consider this and related questions.






share|cite|improve this answer















'Evaluating' $AX_1$, for all $X_1$ will give the column space of $A $, and similarly for $B$. The sum of these two will give the set of vectors $u+v$, wher $u,v $ are elements of column space of $A $ and clmn space of $B $ respectively. That is, we get the (algebraic) sum of the two subspaces (also a subspace). Considering the matrices as operators this is also obviously $Im (A)+Im (B) $.



Sum of such subspaces and similar have been and are studied, and will be considered for example in operator theory. Books on Linear algebra may consider this and related questions.







share|cite|improve this answer















share|cite|improve this answer



share|cite|improve this answer








edited Jul 16 at 0:09


























answered Jul 16 at 0:01









AnyAD

1,451611




1,451611











  • Thanks, @AnyAD. I am not a mathematician, I asked this question as such combination appeared in my research. So, one sub-question, are there any necessary conditions on $A$ and $B$ (like positive definiteness, non-singularity etc) in operator theory?
    – Joy
    Jul 16 at 8:37
















  • Thanks, @AnyAD. I am not a mathematician, I asked this question as such combination appeared in my research. So, one sub-question, are there any necessary conditions on $A$ and $B$ (like positive definiteness, non-singularity etc) in operator theory?
    – Joy
    Jul 16 at 8:37















Thanks, @AnyAD. I am not a mathematician, I asked this question as such combination appeared in my research. So, one sub-question, are there any necessary conditions on $A$ and $B$ (like positive definiteness, non-singularity etc) in operator theory?
– Joy
Jul 16 at 8:37




Thanks, @AnyAD. I am not a mathematician, I asked this question as such combination appeared in my research. So, one sub-question, are there any necessary conditions on $A$ and $B$ (like positive definiteness, non-singularity etc) in operator theory?
– Joy
Jul 16 at 8:37












 

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