are there any relations between these subspaces

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I have a pair of square matrices $(E,A)$ of the same dimension say $n$, consider the following subspaces



First:



$V_0=mathbbR^n, V_i+1=A^-1(EV_i), W_0=0, W_i+1=E^-1(AW_i)$



A matrix $Q$ is called projector onto a subspace $S$ iff $text Im Q=S$ and $Q^2=Q$, It is called a projector along a subspace $S$ iff $ker Q=S$ and $Q^2=Q$.



Second:



we define a matrix chain by setting



$E_0=E$



$ A_0=A,$



$ E_i+1=E_i-A_iQ_i, $



$A_i+1=A_iP_i$



$Q_i$ are are projectors onto $ker E_i, P_i=I-Q_i$



my question or intention is to ask here or maybe someone helps me to convince or point out whether there is or there is no relation between first and second subspaces/ matrices or there is any way to relate them mathematically geometrically slightly changing somewhere in the first or second theme.



In a first way: $V_i$ are decreasing nested chain of subspaces and due to finite dimension, it will be stationary, similar for $W_i$ but they are increasing in nature.



Also, I have proved myself if $det (A-lambda E)ne 0$ then there is some $k$ such that $V_koplus W_k=mathbbR^n, V_kcap W_k=0$.



I have no idea how geometrically looks like the subspaces/matrices in the second case.



if there is some geometric insight hidden there, please help me to understand, so that I myself can relate two, if possible.



Thank you.




linear-algebra matrices vector-spaces linear-transformations




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  • I think you need to learn english
    – miosaki
    Jul 23 at 12:46














up vote
0
down vote

favorite












I have a pair of square matrices $(E,A)$ of the same dimension say $n$, consider the following subspaces



First:



$V_0=mathbbR^n, V_i+1=A^-1(EV_i), W_0=0, W_i+1=E^-1(AW_i)$



A matrix $Q$ is called projector onto a subspace $S$ iff $text Im Q=S$ and $Q^2=Q$, It is called a projector along a subspace $S$ iff $ker Q=S$ and $Q^2=Q$.



Second:



we define a matrix chain by setting



$E_0=E$



$ A_0=A,$



$ E_i+1=E_i-A_iQ_i, $



$A_i+1=A_iP_i$



$Q_i$ are are projectors onto $ker E_i, P_i=I-Q_i$



my question or intention is to ask here or maybe someone helps me to convince or point out whether there is or there is no relation between first and second subspaces/ matrices or there is any way to relate them mathematically geometrically slightly changing somewhere in the first or second theme.



In a first way: $V_i$ are decreasing nested chain of subspaces and due to finite dimension, it will be stationary, similar for $W_i$ but they are increasing in nature.



Also, I have proved myself if $det (A-lambda E)ne 0$ then there is some $k$ such that $V_koplus W_k=mathbbR^n, V_kcap W_k=0$.



I have no idea how geometrically looks like the subspaces/matrices in the second case.



if there is some geometric insight hidden there, please help me to understand, so that I myself can relate two, if possible.



Thank you.




linear-algebra matrices vector-spaces linear-transformations




share|cite|improve this question





















  • I think you need to learn english
    – miosaki
    Jul 23 at 12:46












up vote
0
down vote

favorite









up vote
0
down vote

favorite











I have a pair of square matrices $(E,A)$ of the same dimension say $n$, consider the following subspaces



First:



$V_0=mathbbR^n, V_i+1=A^-1(EV_i), W_0=0, W_i+1=E^-1(AW_i)$



A matrix $Q$ is called projector onto a subspace $S$ iff $text Im Q=S$ and $Q^2=Q$, It is called a projector along a subspace $S$ iff $ker Q=S$ and $Q^2=Q$.



Second:



we define a matrix chain by setting



$E_0=E$



$ A_0=A,$



$ E_i+1=E_i-A_iQ_i, $



$A_i+1=A_iP_i$



$Q_i$ are are projectors onto $ker E_i, P_i=I-Q_i$



my question or intention is to ask here or maybe someone helps me to convince or point out whether there is or there is no relation between first and second subspaces/ matrices or there is any way to relate them mathematically geometrically slightly changing somewhere in the first or second theme.



In a first way: $V_i$ are decreasing nested chain of subspaces and due to finite dimension, it will be stationary, similar for $W_i$ but they are increasing in nature.



Also, I have proved myself if $det (A-lambda E)ne 0$ then there is some $k$ such that $V_koplus W_k=mathbbR^n, V_kcap W_k=0$.



I have no idea how geometrically looks like the subspaces/matrices in the second case.



if there is some geometric insight hidden there, please help me to understand, so that I myself can relate two, if possible.



Thank you.




linear-algebra matrices vector-spaces linear-transformations




share|cite|improve this question













I have a pair of square matrices $(E,A)$ of the same dimension say $n$, consider the following subspaces



First:



$V_0=mathbbR^n, V_i+1=A^-1(EV_i), W_0=0, W_i+1=E^-1(AW_i)$



A matrix $Q$ is called projector onto a subspace $S$ iff $text Im Q=S$ and $Q^2=Q$, It is called a projector along a subspace $S$ iff $ker Q=S$ and $Q^2=Q$.



Second:



we define a matrix chain by setting



$E_0=E$



$ A_0=A,$



$ E_i+1=E_i-A_iQ_i, $



$A_i+1=A_iP_i$



$Q_i$ are are projectors onto $ker E_i, P_i=I-Q_i$



my question or intention is to ask here or maybe someone helps me to convince or point out whether there is or there is no relation between first and second subspaces/ matrices or there is any way to relate them mathematically geometrically slightly changing somewhere in the first or second theme.



In a first way: $V_i$ are decreasing nested chain of subspaces and due to finite dimension, it will be stationary, similar for $W_i$ but they are increasing in nature.



Also, I have proved myself if $det (A-lambda E)ne 0$ then there is some $k$ such that $V_koplus W_k=mathbbR^n, V_kcap W_k=0$.



I have no idea how geometrically looks like the subspaces/matrices in the second case.



if there is some geometric insight hidden there, please help me to understand, so that I myself can relate two, if possible.



Thank you.





linear-algebra matrices vector-spaces linear-transformations





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edited Jul 23 at 10:33
























asked Jul 22 at 10:57









miosaki

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  • I think you need to learn english
    – miosaki
    Jul 23 at 12:46
















  • I think you need to learn english
    – miosaki
    Jul 23 at 12:46















I think you need to learn english
– miosaki
Jul 23 at 12:46




I think you need to learn english
– miosaki
Jul 23 at 12:46















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