Rewriting $left[begin{array}{c} I_notimes e_1\ vdots \ I_n otimes e_T end{array}right]$












0














Is there a way to rewrite the matrix below using operations like $vec$, $otimes$, etc?



$left[begin{array}{c}
I_notimes e_1\
vdots \
I_n otimes e_T end{array}right]$



$e_i$ is the $i$-th column of the matrix $I_T$, $otimes$ is the Kronecker product.



I need to rewrite this matrix to make it faster to work with when programming.






linear-algebra







share|cite|improve this question



























    0














    Is there a way to rewrite the matrix below using operations like $vec$, $otimes$, etc?



    $left[begin{array}{c}
    I_notimes e_1\
    vdots \
    I_n otimes e_T end{array}right]$



    $e_i$ is the $i$-th column of the matrix $I_T$, $otimes$ is the Kronecker product.



    I need to rewrite this matrix to make it faster to work with when programming.






    linear-algebra







    share|cite|improve this question

























      0












      0








      0







      Is there a way to rewrite the matrix below using operations like $vec$, $otimes$, etc?



      $left[begin{array}{c}
      I_notimes e_1\
      vdots \
      I_n otimes e_T end{array}right]$



      $e_i$ is the $i$-th column of the matrix $I_T$, $otimes$ is the Kronecker product.



      I need to rewrite this matrix to make it faster to work with when programming.






      linear-algebra







      share|cite|improve this question













      Is there a way to rewrite the matrix below using operations like $vec$, $otimes$, etc?



      $left[begin{array}{c}
      I_notimes e_1\
      vdots \
      I_n otimes e_T end{array}right]$



      $e_i$ is the $i$-th column of the matrix $I_T$, $otimes$ is the Kronecker product.



      I need to rewrite this matrix to make it faster to work with when programming.






      linear-algebra




      linear-algebra






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Nov 13 '18 at 15:52









      An old man in the sea.

      1,62211032




      1,62211032






















          1 Answer
          1






          active

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          1














          Is your goal to compute matrix-vector products involving this matrix? If this is the case, you could restructure the computation as follows. We have
          $$begin{bmatrix}I_n otimes e_1 newline I_n otimes e_2 newline vdots newline I_n otimes e_Tend{bmatrix} x
          = begin{bmatrix}(I_n otimes e_1)x newline (I_n otimes e_2)x newline vdots newline (I_n otimes e_T)x end{bmatrix}.$$



          Using e.g. equation (2) from the Van Loan paper "The ubiquitous Kronecker product" from 2000, we can write $(I_n otimes e_i) x = text{vec}(e_i x^top)$ for each $i$. We therefore have
          $$begin{bmatrix}I_n otimes e_1 newline I_n otimes e_2 newline vdots newline I_n otimes e_Tend{bmatrix} x =
          begin{bmatrix}
          text{vec}(e_1 x^top) newline text{vec}(e_2 x^top) newline vdots newline text{vec}(e_T x^top)
          end{bmatrix}.$$
          This avoids having to form the full Kronecker product.



          That Van Loan paper I cited above contains lots of useful information about Kronecker products if you deal with them a lot.






          share|cite|improve this answer





















          • Thanks OtZman ;)
            – An old man in the sea.
            Dec 21 '18 at 8:41










          • If you're insterested ;) math.stackexchange.com/questions/3048304/…
            – An old man in the sea.
            Dec 21 '18 at 8:45











          Your Answer





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          1 Answer
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          active

          oldest

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          1 Answer
          1






          active

          oldest

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          active

          oldest

          votes






          active

          oldest

          votes









          1














          Is your goal to compute matrix-vector products involving this matrix? If this is the case, you could restructure the computation as follows. We have
          $$begin{bmatrix}I_n otimes e_1 newline I_n otimes e_2 newline vdots newline I_n otimes e_Tend{bmatrix} x
          = begin{bmatrix}(I_n otimes e_1)x newline (I_n otimes e_2)x newline vdots newline (I_n otimes e_T)x end{bmatrix}.$$



          Using e.g. equation (2) from the Van Loan paper "The ubiquitous Kronecker product" from 2000, we can write $(I_n otimes e_i) x = text{vec}(e_i x^top)$ for each $i$. We therefore have
          $$begin{bmatrix}I_n otimes e_1 newline I_n otimes e_2 newline vdots newline I_n otimes e_Tend{bmatrix} x =
          begin{bmatrix}
          text{vec}(e_1 x^top) newline text{vec}(e_2 x^top) newline vdots newline text{vec}(e_T x^top)
          end{bmatrix}.$$
          This avoids having to form the full Kronecker product.



          That Van Loan paper I cited above contains lots of useful information about Kronecker products if you deal with them a lot.






          share|cite|improve this answer





















          • Thanks OtZman ;)
            – An old man in the sea.
            Dec 21 '18 at 8:41










          • If you're insterested ;) math.stackexchange.com/questions/3048304/…
            – An old man in the sea.
            Dec 21 '18 at 8:45
















          1














          Is your goal to compute matrix-vector products involving this matrix? If this is the case, you could restructure the computation as follows. We have
          $$begin{bmatrix}I_n otimes e_1 newline I_n otimes e_2 newline vdots newline I_n otimes e_Tend{bmatrix} x
          = begin{bmatrix}(I_n otimes e_1)x newline (I_n otimes e_2)x newline vdots newline (I_n otimes e_T)x end{bmatrix}.$$



          Using e.g. equation (2) from the Van Loan paper "The ubiquitous Kronecker product" from 2000, we can write $(I_n otimes e_i) x = text{vec}(e_i x^top)$ for each $i$. We therefore have
          $$begin{bmatrix}I_n otimes e_1 newline I_n otimes e_2 newline vdots newline I_n otimes e_Tend{bmatrix} x =
          begin{bmatrix}
          text{vec}(e_1 x^top) newline text{vec}(e_2 x^top) newline vdots newline text{vec}(e_T x^top)
          end{bmatrix}.$$
          This avoids having to form the full Kronecker product.



          That Van Loan paper I cited above contains lots of useful information about Kronecker products if you deal with them a lot.






          share|cite|improve this answer





















          • Thanks OtZman ;)
            – An old man in the sea.
            Dec 21 '18 at 8:41










          • If you're insterested ;) math.stackexchange.com/questions/3048304/…
            – An old man in the sea.
            Dec 21 '18 at 8:45














          1












          1








          1






          Is your goal to compute matrix-vector products involving this matrix? If this is the case, you could restructure the computation as follows. We have
          $$begin{bmatrix}I_n otimes e_1 newline I_n otimes e_2 newline vdots newline I_n otimes e_Tend{bmatrix} x
          = begin{bmatrix}(I_n otimes e_1)x newline (I_n otimes e_2)x newline vdots newline (I_n otimes e_T)x end{bmatrix}.$$



          Using e.g. equation (2) from the Van Loan paper "The ubiquitous Kronecker product" from 2000, we can write $(I_n otimes e_i) x = text{vec}(e_i x^top)$ for each $i$. We therefore have
          $$begin{bmatrix}I_n otimes e_1 newline I_n otimes e_2 newline vdots newline I_n otimes e_Tend{bmatrix} x =
          begin{bmatrix}
          text{vec}(e_1 x^top) newline text{vec}(e_2 x^top) newline vdots newline text{vec}(e_T x^top)
          end{bmatrix}.$$
          This avoids having to form the full Kronecker product.



          That Van Loan paper I cited above contains lots of useful information about Kronecker products if you deal with them a lot.






          share|cite|improve this answer












          Is your goal to compute matrix-vector products involving this matrix? If this is the case, you could restructure the computation as follows. We have
          $$begin{bmatrix}I_n otimes e_1 newline I_n otimes e_2 newline vdots newline I_n otimes e_Tend{bmatrix} x
          = begin{bmatrix}(I_n otimes e_1)x newline (I_n otimes e_2)x newline vdots newline (I_n otimes e_T)x end{bmatrix}.$$



          Using e.g. equation (2) from the Van Loan paper "The ubiquitous Kronecker product" from 2000, we can write $(I_n otimes e_i) x = text{vec}(e_i x^top)$ for each $i$. We therefore have
          $$begin{bmatrix}I_n otimes e_1 newline I_n otimes e_2 newline vdots newline I_n otimes e_Tend{bmatrix} x =
          begin{bmatrix}
          text{vec}(e_1 x^top) newline text{vec}(e_2 x^top) newline vdots newline text{vec}(e_T x^top)
          end{bmatrix}.$$
          This avoids having to form the full Kronecker product.



          That Van Loan paper I cited above contains lots of useful information about Kronecker products if you deal with them a lot.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Nov 19 '18 at 3:35









          OtZman

          814




          814












          • Thanks OtZman ;)
            – An old man in the sea.
            Dec 21 '18 at 8:41










          • If you're insterested ;) math.stackexchange.com/questions/3048304/…
            – An old man in the sea.
            Dec 21 '18 at 8:45


















          • Thanks OtZman ;)
            – An old man in the sea.
            Dec 21 '18 at 8:41










          • If you're insterested ;) math.stackexchange.com/questions/3048304/…
            – An old man in the sea.
            Dec 21 '18 at 8:45
















          Thanks OtZman ;)
          – An old man in the sea.
          Dec 21 '18 at 8:41




          Thanks OtZman ;)
          – An old man in the sea.
          Dec 21 '18 at 8:41












          If you're insterested ;) math.stackexchange.com/questions/3048304/…
          – An old man in the sea.
          Dec 21 '18 at 8:45




          If you're insterested ;) math.stackexchange.com/questions/3048304/…
          – An old man in the sea.
          Dec 21 '18 at 8:45


















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