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Question on elementary Linear Algebra product

4

$begingroup$

I came across this in a problem:

$$fracmathbfumathbfv^TmathbfA^-1+mathbfumathbfv^TmathbfA^-1mathbfumathbfv^TmathbfA^-11+mathbfv^TmathbfA^-1mathbfu$$

Then, in order to simplify this, I thought the following:

begingather
fracmathbfumathbfv^TmathbfA^-1+mathbfucolorredleft(mathbfv^TmathbfA^-1mathbfuright)mathbfv^TmathbfA^-11+mathbfv^TmathbfA^-1mathbfu\
fracmathbfumathbfv^TmathbfA^-1+mathbfumathbfv^TmathbfA^-1colorredleft(mathbfv^TmathbfA^-1mathbfuright)1+mathbfv^TmathbfA^-1mathbfu\
fracmathbfumathbfv^TmathbfA^-1left(1+colorredmathbfv^TmathbfA^-1mathbfuright)1+mathbfv^TmathbfA^-1mathbfu\
mathbfumathbfv^TmathbfA^-1
endgather

I figured that this could be done because of associativity and as the terms in $colorredtextred$ result in a scalar, it could be 'moved around'. Is this correct? If yes, does this always hold or are there any conditions on the vectors and matrices involved?

matrices vectors

share|cite|improve this question

asked Mar 26 at 6:16

bertozzijrbertozzijr

6331720

$endgroup$

• 
  
  
  

  
  4
  

  
  
  
  
  
  

  
  $begingroup$
  Correct, and it always holds (as long as $A$ is invertible of course).
  $endgroup$
  – amsmath
  Mar 26 at 6:21
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @amsmath great, thx!
  $endgroup$
  – bertozzijr
  Mar 26 at 7:34
  

  
  
  
  

add a comment | 

4

$begingroup$

I came across this in a problem:

$$fracmathbfumathbfv^TmathbfA^-1+mathbfumathbfv^TmathbfA^-1mathbfumathbfv^TmathbfA^-11+mathbfv^TmathbfA^-1mathbfu$$

Then, in order to simplify this, I thought the following:

begingather
fracmathbfumathbfv^TmathbfA^-1+mathbfucolorredleft(mathbfv^TmathbfA^-1mathbfuright)mathbfv^TmathbfA^-11+mathbfv^TmathbfA^-1mathbfu\
fracmathbfumathbfv^TmathbfA^-1+mathbfumathbfv^TmathbfA^-1colorredleft(mathbfv^TmathbfA^-1mathbfuright)1+mathbfv^TmathbfA^-1mathbfu\
fracmathbfumathbfv^TmathbfA^-1left(1+colorredmathbfv^TmathbfA^-1mathbfuright)1+mathbfv^TmathbfA^-1mathbfu\
mathbfumathbfv^TmathbfA^-1
endgather

I figured that this could be done because of associativity and as the terms in $colorredtextred$ result in a scalar, it could be 'moved around'. Is this correct? If yes, does this always hold or are there any conditions on the vectors and matrices involved?

matrices vectors

share|cite|improve this question

asked Mar 26 at 6:16

bertozzijrbertozzijr

6331720

$endgroup$

• 
  
  
  

  
  4
  

  
  
  
  
  
  

  
  $begingroup$
  Correct, and it always holds (as long as $A$ is invertible of course).
  $endgroup$
  – amsmath
  Mar 26 at 6:21
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @amsmath great, thx!
  $endgroup$
  – bertozzijr
  Mar 26 at 7:34
  

  
  
  
  

add a comment | 

$begingroup$

I came across this in a problem:

$$fracmathbfumathbfv^TmathbfA^-1+mathbfumathbfv^TmathbfA^-1mathbfumathbfv^TmathbfA^-11+mathbfv^TmathbfA^-1mathbfu$$

Then, in order to simplify this, I thought the following:

begingather
fracmathbfumathbfv^TmathbfA^-1+mathbfucolorredleft(mathbfv^TmathbfA^-1mathbfuright)mathbfv^TmathbfA^-11+mathbfv^TmathbfA^-1mathbfu\
fracmathbfumathbfv^TmathbfA^-1+mathbfumathbfv^TmathbfA^-1colorredleft(mathbfv^TmathbfA^-1mathbfuright)1+mathbfv^TmathbfA^-1mathbfu\
fracmathbfumathbfv^TmathbfA^-1left(1+colorredmathbfv^TmathbfA^-1mathbfuright)1+mathbfv^TmathbfA^-1mathbfu\
mathbfumathbfv^TmathbfA^-1
endgather

I figured that this could be done because of associativity and as the terms in $colorredtextred$ result in a scalar, it could be 'moved around'. Is this correct? If yes, does this always hold or are there any conditions on the vectors and matrices involved?

matrices vectors

share|cite|improve this question

asked Mar 26 at 6:16

bertozzijrbertozzijr

6331720

$endgroup$

share|cite|improve this question

asked Mar 26 at 6:16

bertozzijrbertozzijr

6331720

share|cite|improve this question

asked Mar 26 at 6:16

bertozzijrbertozzijr

6331720

asked Mar 26 at 6:16

bertozzijrbertozzijr

6331720

asked Mar 26 at 6:16

bertozzijrbertozzijr

6331720