Simplification of $x = left(A + sum_i alpha_i B_iright)^-1 A y$, $A succ 0, A in M_n,n(mathbbC)$, $textrmrank(B_i) = 1$

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I have a matrix inversion problem on hand. I want to reduce the matrix inversion complexity, if at all feasible. Let me give a brief overview of the problem definition.



Problem definition:
Say $A succ 0, A in M_n,nleft( mathbbC right)$, is Hermitian and full-rank (positive definite) matrix. $B_i in M_n,nleft( mathbbC right)$ and rank-$1$, $alpha_i in mathbbC$ some constant, $y in M_n,1left( mathbbC right)$ some column vector.



beginalign
x = left(A + sum_i=0^m alpha_i B_iright)^-1 A y
endalign



Question: Can this matrix inversion be simplified such that big matrix inversion can be avoided (since $n approx 8$k)? Thank you in advance.



I can't make any progress with matrix inversion lemmas, e.g., in matrix cookbook.




linear-algebra matrices inverse




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  • 1




    does this (math.stackexchange.com/questions/17776/…) help you at all?
    – bobcliffe
    Aug 23 at 9:01











  • Thanks, I will try that.
    – user550103
    Aug 23 at 9:10














up vote
0
down vote

favorite












I have a matrix inversion problem on hand. I want to reduce the matrix inversion complexity, if at all feasible. Let me give a brief overview of the problem definition.



Problem definition:
Say $A succ 0, A in M_n,nleft( mathbbC right)$, is Hermitian and full-rank (positive definite) matrix. $B_i in M_n,nleft( mathbbC right)$ and rank-$1$, $alpha_i in mathbbC$ some constant, $y in M_n,1left( mathbbC right)$ some column vector.



beginalign
x = left(A + sum_i=0^m alpha_i B_iright)^-1 A y
endalign



Question: Can this matrix inversion be simplified such that big matrix inversion can be avoided (since $n approx 8$k)? Thank you in advance.



I can't make any progress with matrix inversion lemmas, e.g., in matrix cookbook.




linear-algebra matrices inverse




share|cite|improve this question
















  • 1




    does this (math.stackexchange.com/questions/17776/…) help you at all?
    – bobcliffe
    Aug 23 at 9:01











  • Thanks, I will try that.
    – user550103
    Aug 23 at 9:10












up vote
0
down vote

favorite









up vote
0
down vote

favorite











I have a matrix inversion problem on hand. I want to reduce the matrix inversion complexity, if at all feasible. Let me give a brief overview of the problem definition.



Problem definition:
Say $A succ 0, A in M_n,nleft( mathbbC right)$, is Hermitian and full-rank (positive definite) matrix. $B_i in M_n,nleft( mathbbC right)$ and rank-$1$, $alpha_i in mathbbC$ some constant, $y in M_n,1left( mathbbC right)$ some column vector.



beginalign
x = left(A + sum_i=0^m alpha_i B_iright)^-1 A y
endalign



Question: Can this matrix inversion be simplified such that big matrix inversion can be avoided (since $n approx 8$k)? Thank you in advance.



I can't make any progress with matrix inversion lemmas, e.g., in matrix cookbook.




linear-algebra matrices inverse




share|cite|improve this question












I have a matrix inversion problem on hand. I want to reduce the matrix inversion complexity, if at all feasible. Let me give a brief overview of the problem definition.



Problem definition:
Say $A succ 0, A in M_n,nleft( mathbbC right)$, is Hermitian and full-rank (positive definite) matrix. $B_i in M_n,nleft( mathbbC right)$ and rank-$1$, $alpha_i in mathbbC$ some constant, $y in M_n,1left( mathbbC right)$ some column vector.



beginalign
x = left(A + sum_i=0^m alpha_i B_iright)^-1 A y
endalign



Question: Can this matrix inversion be simplified such that big matrix inversion can be avoided (since $n approx 8$k)? Thank you in advance.



I can't make any progress with matrix inversion lemmas, e.g., in matrix cookbook.





linear-algebra matrices inverse





share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Aug 23 at 8:29









user550103

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  • 1




    does this (math.stackexchange.com/questions/17776/…) help you at all?
    – bobcliffe
    Aug 23 at 9:01











  • Thanks, I will try that.
    – user550103
    Aug 23 at 9:10












  • 1




    does this (math.stackexchange.com/questions/17776/…) help you at all?
    – bobcliffe
    Aug 23 at 9:01











  • Thanks, I will try that.
    – user550103
    Aug 23 at 9:10







1




1




does this (math.stackexchange.com/questions/17776/…) help you at all?
– bobcliffe
Aug 23 at 9:01





does this (math.stackexchange.com/questions/17776/…) help you at all?
– bobcliffe
Aug 23 at 9:01













Thanks, I will try that.
– user550103
Aug 23 at 9:10




Thanks, I will try that.
– user550103
Aug 23 at 9:10















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