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Taking the conjugate of $(x_j e_j A)$

Characterizing orthonormal basis of I-c v v'Is there any standard notation for specifying dimension of a matrix after the matrix symbol?Very basic Lie Algebras question on the complex conjugate of the adjoint mapShowing $A_ij = (Ae_i, e_j)$ for matrix $A$ of complex linear operator $mathbbC^n to mathbbC^n$ and orthonormal basis $(e_i)_i=1^n$Which matrices commute with A symmetric positive-definite?If $sum_j overlineE_j AE_j^dagger=sum_j E_j AE_j^dagger$ for all $A$, does $ overlineE_j AE_j^dagger= E_j AE_j^dagger$If $sum_j overlineE_j AE_j^dagger=sum_j E_j AE_j^dagger$ for all $A$ what are the conditions on the sequence $E_j$?Show that the matrix respect to $T^*$ is the same as the conjugate transpose of the matrix of $T$Verifying the row-rank and column-rank of a matrix are equal by finding bases for eachConjugate symmetry of an inner product

0

$begingroup$

Let $k in (mathbbR, mathbbC, mathbbH)$ and $A in M_n (k)$, i.e, the set of $n times n$ matrices. Let $e_j = (0,...,0,1,0,...,0)$ an element of the standard orthonormal basis, and $x_j in k$. I'm trying to take the conjugate of the expression:

beginequation*
x_j e_j A
endequation*

I know that for any $x, y in k$, $overlinexy = bary barx$, however I'm not quite sure that $overlinex_j e_j A = overlinee_j A overline x_j$ since $e_jA$ is a column matrix and $x_j$ is a row matrix thus $overlinee_j A overline x_j$ produces a $n times n$ matrix whereas $overlinex_j e_j A$ is a row matrix.

So how would I take the conjugate of $x_j e_j A$?

linear-algebra matrices

share|cite|improve this question

asked Mar 16 at 10:40

XenidiaXenidia

1,285832

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  You said $x_j in k$. That makes it a scalar, not a row matrix. And since $e_j$ is a row matrix, $e_jA$ is also a row matrix. (If $e_j$ were a column matrix, then the expression $e_jA$ makes no sense, as the dimensions do not line up. $A$ multiplies column matrices on left, and row matrices on the right.)
  $endgroup$
  – Paul Sinclair
  Mar 16 at 20:01
  

  
  
  
  

add a comment | 

0

$begingroup$

Let $k in (mathbbR, mathbbC, mathbbH)$ and $A in M_n (k)$, i.e, the set of $n times n$ matrices. Let $e_j = (0,...,0,1,0,...,0)$ an element of the standard orthonormal basis, and $x_j in k$. I'm trying to take the conjugate of the expression:

beginequation*
x_j e_j A
endequation*

I know that for any $x, y in k$, $overlinexy = bary barx$, however I'm not quite sure that $overlinex_j e_j A = overlinee_j A overline x_j$ since $e_jA$ is a column matrix and $x_j$ is a row matrix thus $overlinee_j A overline x_j$ produces a $n times n$ matrix whereas $overlinex_j e_j A$ is a row matrix.

So how would I take the conjugate of $x_j e_j A$?

linear-algebra matrices

share|cite|improve this question

asked Mar 16 at 10:40

XenidiaXenidia

1,285832

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  You said $x_j in k$. That makes it a scalar, not a row matrix. And since $e_j$ is a row matrix, $e_jA$ is also a row matrix. (If $e_j$ were a column matrix, then the expression $e_jA$ makes no sense, as the dimensions do not line up. $A$ multiplies column matrices on left, and row matrices on the right.)
  $endgroup$
  – Paul Sinclair
  Mar 16 at 20:01
  

  
  
  
  

add a comment | 

$begingroup$

Let $k in (mathbbR, mathbbC, mathbbH)$ and $A in M_n (k)$, i.e, the set of $n times n$ matrices. Let $e_j = (0,...,0,1,0,...,0)$ an element of the standard orthonormal basis, and $x_j in k$. I'm trying to take the conjugate of the expression:

beginequation*
x_j e_j A
endequation*

I know that for any $x, y in k$, $overlinexy = bary barx$, however I'm not quite sure that $overlinex_j e_j A = overlinee_j A overline x_j$ since $e_jA$ is a column matrix and $x_j$ is a row matrix thus $overlinee_j A overline x_j$ produces a $n times n$ matrix whereas $overlinex_j e_j A$ is a row matrix.

So how would I take the conjugate of $x_j e_j A$?

linear-algebra matrices

share|cite|improve this question

asked Mar 16 at 10:40

XenidiaXenidia

1,285832

$endgroup$

share|cite|improve this question

asked Mar 16 at 10:40

XenidiaXenidia

1,285832

share|cite|improve this question

asked Mar 16 at 10:40

XenidiaXenidia

1,285832

asked Mar 16 at 10:40

XenidiaXenidia

1,285832

asked Mar 16 at 10:40

XenidiaXenidia

1,285832