Do matrix multiplication rules apply when multiplying matrices made up of smaller matrices?












3












$begingroup$


I came across a proof that did the following.



$a$, $b$, $c$ are $3 times 1$ vectors. $A$ is a $3 times 3$ matrix and $d$ is a $3 times 1$ vector



$begin{bmatrix}A & dend{bmatrix} begin{bmatrix} a & b & c \ 0 & 0 & 1 end{bmatrix} = begin{bmatrix} Aa & Ab & Ac +dend{bmatrix}$



So we are left multiplying a $4 times 3$ matrix by a $3 times 4$ matrix, but the elements of both matrices are themselves vectors and matrices.



I know this is correct because the rest of the proof in the paper follows. But, I don't really know how to justify using regular matrix multiplication properties to multiply two matrices made up of matrices. For example, $Aa$ is a matrix-vector product, whereas in regular matrix multiplication it would be some scalar product.




  1. Is there a name for this kind of linear algebra property & are there more like it?


  2. Why does it work? (Just a reference would suffice)







linear-algebra







share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    You can try reading about block matrices and their arithmetic operations.
    $endgroup$
    – Shivani Goel
    Dec 17 '18 at 10:31






  • 4




    $begingroup$
    en.wikipedia.org/wiki/Block_matrix
    $endgroup$
    – Rahul
    Dec 17 '18 at 10:33










  • $begingroup$
    Thank you both! I didn't know there was something called Block Matrix.
    $endgroup$
    – Carpetfizz
    Dec 17 '18 at 10:34






  • 1




    $begingroup$
    I'm fairly sure that matrix multiplication is thoroughly recursive in that respect. Don't rely on what I've said though ... I just have a recollection of learning that it is.
    $endgroup$
    – AmbretteOrrisey
    Dec 17 '18 at 11:00
















3












$begingroup$


I came across a proof that did the following.



$a$, $b$, $c$ are $3 times 1$ vectors. $A$ is a $3 times 3$ matrix and $d$ is a $3 times 1$ vector



$begin{bmatrix}A & dend{bmatrix} begin{bmatrix} a & b & c \ 0 & 0 & 1 end{bmatrix} = begin{bmatrix} Aa & Ab & Ac +dend{bmatrix}$



So we are left multiplying a $4 times 3$ matrix by a $3 times 4$ matrix, but the elements of both matrices are themselves vectors and matrices.



I know this is correct because the rest of the proof in the paper follows. But, I don't really know how to justify using regular matrix multiplication properties to multiply two matrices made up of matrices. For example, $Aa$ is a matrix-vector product, whereas in regular matrix multiplication it would be some scalar product.




  1. Is there a name for this kind of linear algebra property & are there more like it?


  2. Why does it work? (Just a reference would suffice)







linear-algebra







share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    You can try reading about block matrices and their arithmetic operations.
    $endgroup$
    – Shivani Goel
    Dec 17 '18 at 10:31






  • 4




    $begingroup$
    en.wikipedia.org/wiki/Block_matrix
    $endgroup$
    – Rahul
    Dec 17 '18 at 10:33










  • $begingroup$
    Thank you both! I didn't know there was something called Block Matrix.
    $endgroup$
    – Carpetfizz
    Dec 17 '18 at 10:34






  • 1




    $begingroup$
    I'm fairly sure that matrix multiplication is thoroughly recursive in that respect. Don't rely on what I've said though ... I just have a recollection of learning that it is.
    $endgroup$
    – AmbretteOrrisey
    Dec 17 '18 at 11:00














3












3








3


1



$begingroup$


I came across a proof that did the following.



$a$, $b$, $c$ are $3 times 1$ vectors. $A$ is a $3 times 3$ matrix and $d$ is a $3 times 1$ vector



$begin{bmatrix}A & dend{bmatrix} begin{bmatrix} a & b & c \ 0 & 0 & 1 end{bmatrix} = begin{bmatrix} Aa & Ab & Ac +dend{bmatrix}$



So we are left multiplying a $4 times 3$ matrix by a $3 times 4$ matrix, but the elements of both matrices are themselves vectors and matrices.



I know this is correct because the rest of the proof in the paper follows. But, I don't really know how to justify using regular matrix multiplication properties to multiply two matrices made up of matrices. For example, $Aa$ is a matrix-vector product, whereas in regular matrix multiplication it would be some scalar product.




  1. Is there a name for this kind of linear algebra property & are there more like it?


  2. Why does it work? (Just a reference would suffice)







linear-algebra







share|cite|improve this question











$endgroup$




I came across a proof that did the following.



$a$, $b$, $c$ are $3 times 1$ vectors. $A$ is a $3 times 3$ matrix and $d$ is a $3 times 1$ vector



$begin{bmatrix}A & dend{bmatrix} begin{bmatrix} a & b & c \ 0 & 0 & 1 end{bmatrix} = begin{bmatrix} Aa & Ab & Ac +dend{bmatrix}$



So we are left multiplying a $4 times 3$ matrix by a $3 times 4$ matrix, but the elements of both matrices are themselves vectors and matrices.



I know this is correct because the rest of the proof in the paper follows. But, I don't really know how to justify using regular matrix multiplication properties to multiply two matrices made up of matrices. For example, $Aa$ is a matrix-vector product, whereas in regular matrix multiplication it would be some scalar product.




  1. Is there a name for this kind of linear algebra property & are there more like it?


  2. Why does it work? (Just a reference would suffice)







linear-algebra




linear-algebra






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 17 '18 at 10:56









The Pointer

2,64421638




2,64421638










asked Dec 17 '18 at 10:24









CarpetfizzCarpetfizz

491413




491413








  • 1




    $begingroup$
    You can try reading about block matrices and their arithmetic operations.
    $endgroup$
    – Shivani Goel
    Dec 17 '18 at 10:31






  • 4




    $begingroup$
    en.wikipedia.org/wiki/Block_matrix
    $endgroup$
    – Rahul
    Dec 17 '18 at 10:33










  • $begingroup$
    Thank you both! I didn't know there was something called Block Matrix.
    $endgroup$
    – Carpetfizz
    Dec 17 '18 at 10:34






  • 1




    $begingroup$
    I'm fairly sure that matrix multiplication is thoroughly recursive in that respect. Don't rely on what I've said though ... I just have a recollection of learning that it is.
    $endgroup$
    – AmbretteOrrisey
    Dec 17 '18 at 11:00














  • 1




    $begingroup$
    You can try reading about block matrices and their arithmetic operations.
    $endgroup$
    – Shivani Goel
    Dec 17 '18 at 10:31






  • 4




    $begingroup$
    en.wikipedia.org/wiki/Block_matrix
    $endgroup$
    – Rahul
    Dec 17 '18 at 10:33










  • $begingroup$
    Thank you both! I didn't know there was something called Block Matrix.
    $endgroup$
    – Carpetfizz
    Dec 17 '18 at 10:34






  • 1




    $begingroup$
    I'm fairly sure that matrix multiplication is thoroughly recursive in that respect. Don't rely on what I've said though ... I just have a recollection of learning that it is.
    $endgroup$
    – AmbretteOrrisey
    Dec 17 '18 at 11:00








1




1




$begingroup$
You can try reading about block matrices and their arithmetic operations.
$endgroup$
– Shivani Goel
Dec 17 '18 at 10:31




$begingroup$
You can try reading about block matrices and their arithmetic operations.
$endgroup$
– Shivani Goel
Dec 17 '18 at 10:31




4




4




$begingroup$
en.wikipedia.org/wiki/Block_matrix
$endgroup$
– Rahul
Dec 17 '18 at 10:33




$begingroup$
en.wikipedia.org/wiki/Block_matrix
$endgroup$
– Rahul
Dec 17 '18 at 10:33












$begingroup$
Thank you both! I didn't know there was something called Block Matrix.
$endgroup$
– Carpetfizz
Dec 17 '18 at 10:34




$begingroup$
Thank you both! I didn't know there was something called Block Matrix.
$endgroup$
– Carpetfizz
Dec 17 '18 at 10:34




1




1




$begingroup$
I'm fairly sure that matrix multiplication is thoroughly recursive in that respect. Don't rely on what I've said though ... I just have a recollection of learning that it is.
$endgroup$
– AmbretteOrrisey
Dec 17 '18 at 11:00




$begingroup$
I'm fairly sure that matrix multiplication is thoroughly recursive in that respect. Don't rely on what I've said though ... I just have a recollection of learning that it is.
$endgroup$
– AmbretteOrrisey
Dec 17 '18 at 11:00










1 Answer
1






active

oldest

votes


















6












$begingroup$

This block matrix notation is very useful. The meaning of the notation is clear: for example, if
$$
A = begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} end{bmatrix}, quad
B = begin{bmatrix} b_{11} & b_{12} \ b_{21} & b_{22} end{bmatrix}, quad
C = begin{bmatrix} c_{11} & c_{12} \ c_{21} & c_{22} end{bmatrix}, quad
D = begin{bmatrix} d_{11} & d_{12} \ d_{21} & d_{22} end{bmatrix}
$$
then $begin{bmatrix} A & B \ C & D end{bmatrix}$ denotes the $4 times 4$ matrix
$$
begin{bmatrix} A & B \ C & D end{bmatrix} =
begin{bmatrix} a_{11} & a_{12} & b_{11} & b_{12} \
a_{21} & a_{22} & b_{21} & b_{22} \
c_{11} & c_{12} & d_{11} & d_{12} \
c_{21} & c_{22} & d_{21} & d_{22}
end{bmatrix}.
$$
It is straightforward to prove the following basic rules
for matrix multiplication using block notation:





  • $A begin{bmatrix} B & C end{bmatrix} =
    begin{bmatrix} AB & AC end{bmatrix}$    .

  • $begin{bmatrix}
    A \
    B
    end{bmatrix} C =
    begin{bmatrix}
    AC \
    BC
    end{bmatrix}.
    $

  • $
    begin{bmatrix} A & B end{bmatrix} begin{bmatrix} C \ D end{bmatrix}
    = AC + BD.
    $


(For each rule, we must assume that
the matrices $A, B, C$, and $D$
have compatible shapes.)



Using these basic rules, we can easily derive any more complicated block matrix multiplication rule that we need. For example,



begin{align}
begin{bmatrix} A & B \ C & D end{bmatrix} begin{bmatrix} X \ Y end{bmatrix}
&=
begin{bmatrix}
begin{bmatrix} A & B end{bmatrix} begin{bmatrix} X \ Y end{bmatrix} \
begin{bmatrix} C & D end{bmatrix} begin{bmatrix} X \ Y end{bmatrix}
end{bmatrix} \
&=
begin{bmatrix}
AX + BY \
CX + DY
end{bmatrix}
end{align}
(assuming that the matrices $A, B, C, D, X$, and $Y$ have compatible shapes).






share|cite|improve this answer











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    active

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    6












    $begingroup$

    This block matrix notation is very useful. The meaning of the notation is clear: for example, if
    $$
    A = begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} end{bmatrix}, quad
    B = begin{bmatrix} b_{11} & b_{12} \ b_{21} & b_{22} end{bmatrix}, quad
    C = begin{bmatrix} c_{11} & c_{12} \ c_{21} & c_{22} end{bmatrix}, quad
    D = begin{bmatrix} d_{11} & d_{12} \ d_{21} & d_{22} end{bmatrix}
    $$
    then $begin{bmatrix} A & B \ C & D end{bmatrix}$ denotes the $4 times 4$ matrix
    $$
    begin{bmatrix} A & B \ C & D end{bmatrix} =
    begin{bmatrix} a_{11} & a_{12} & b_{11} & b_{12} \
    a_{21} & a_{22} & b_{21} & b_{22} \
    c_{11} & c_{12} & d_{11} & d_{12} \
    c_{21} & c_{22} & d_{21} & d_{22}
    end{bmatrix}.
    $$
    It is straightforward to prove the following basic rules
    for matrix multiplication using block notation:





    • $A begin{bmatrix} B & C end{bmatrix} =
      begin{bmatrix} AB & AC end{bmatrix}$      .

    • $begin{bmatrix}
      A \
      B
      end{bmatrix} C =
      begin{bmatrix}
      AC \
      BC
      end{bmatrix}.
      $

    • $
      begin{bmatrix} A & B end{bmatrix} begin{bmatrix} C \ D end{bmatrix}
      = AC + BD.
      $


    (For each rule, we must assume that
    the matrices $A, B, C$, and $D$
    have compatible shapes.)



    Using these basic rules, we can easily derive any more complicated block matrix multiplication rule that we need. For example,



    begin{align}
    begin{bmatrix} A & B \ C & D end{bmatrix} begin{bmatrix} X \ Y end{bmatrix}
    &=
    begin{bmatrix}
    begin{bmatrix} A & B end{bmatrix} begin{bmatrix} X \ Y end{bmatrix} \
    begin{bmatrix} C & D end{bmatrix} begin{bmatrix} X \ Y end{bmatrix}
    end{bmatrix} \
    &=
    begin{bmatrix}
    AX + BY \
    CX + DY
    end{bmatrix}
    end{align}
    (assuming that the matrices $A, B, C, D, X$, and $Y$ have compatible shapes).






    share|cite|improve this answer











    $endgroup$


















      6












      $begingroup$

      This block matrix notation is very useful. The meaning of the notation is clear: for example, if
      $$
      A = begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} end{bmatrix}, quad
      B = begin{bmatrix} b_{11} & b_{12} \ b_{21} & b_{22} end{bmatrix}, quad
      C = begin{bmatrix} c_{11} & c_{12} \ c_{21} & c_{22} end{bmatrix}, quad
      D = begin{bmatrix} d_{11} & d_{12} \ d_{21} & d_{22} end{bmatrix}
      $$
      then $begin{bmatrix} A & B \ C & D end{bmatrix}$ denotes the $4 times 4$ matrix
      $$
      begin{bmatrix} A & B \ C & D end{bmatrix} =
      begin{bmatrix} a_{11} & a_{12} & b_{11} & b_{12} \
      a_{21} & a_{22} & b_{21} & b_{22} \
      c_{11} & c_{12} & d_{11} & d_{12} \
      c_{21} & c_{22} & d_{21} & d_{22}
      end{bmatrix}.
      $$
      It is straightforward to prove the following basic rules
      for matrix multiplication using block notation:





      • $A begin{bmatrix} B & C end{bmatrix} =
        begin{bmatrix} AB & AC end{bmatrix}$        .

      • $begin{bmatrix}
        A \
        B
        end{bmatrix} C =
        begin{bmatrix}
        AC \
        BC
        end{bmatrix}.
        $

      • $
        begin{bmatrix} A & B end{bmatrix} begin{bmatrix} C \ D end{bmatrix}
        = AC + BD.
        $


      (For each rule, we must assume that
      the matrices $A, B, C$, and $D$
      have compatible shapes.)



      Using these basic rules, we can easily derive any more complicated block matrix multiplication rule that we need. For example,



      begin{align}
      begin{bmatrix} A & B \ C & D end{bmatrix} begin{bmatrix} X \ Y end{bmatrix}
      &=
      begin{bmatrix}
      begin{bmatrix} A & B end{bmatrix} begin{bmatrix} X \ Y end{bmatrix} \
      begin{bmatrix} C & D end{bmatrix} begin{bmatrix} X \ Y end{bmatrix}
      end{bmatrix} \
      &=
      begin{bmatrix}
      AX + BY \
      CX + DY
      end{bmatrix}
      end{align}
      (assuming that the matrices $A, B, C, D, X$, and $Y$ have compatible shapes).






      share|cite|improve this answer











      $endgroup$
















        6












        6








        6





        $begingroup$

        This block matrix notation is very useful. The meaning of the notation is clear: for example, if
        $$
        A = begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} end{bmatrix}, quad
        B = begin{bmatrix} b_{11} & b_{12} \ b_{21} & b_{22} end{bmatrix}, quad
        C = begin{bmatrix} c_{11} & c_{12} \ c_{21} & c_{22} end{bmatrix}, quad
        D = begin{bmatrix} d_{11} & d_{12} \ d_{21} & d_{22} end{bmatrix}
        $$
        then $begin{bmatrix} A & B \ C & D end{bmatrix}$ denotes the $4 times 4$ matrix
        $$
        begin{bmatrix} A & B \ C & D end{bmatrix} =
        begin{bmatrix} a_{11} & a_{12} & b_{11} & b_{12} \
        a_{21} & a_{22} & b_{21} & b_{22} \
        c_{11} & c_{12} & d_{11} & d_{12} \
        c_{21} & c_{22} & d_{21} & d_{22}
        end{bmatrix}.
        $$
        It is straightforward to prove the following basic rules
        for matrix multiplication using block notation:





        • $A begin{bmatrix} B & C end{bmatrix} =
          begin{bmatrix} AB & AC end{bmatrix}$          .

        • $begin{bmatrix}
          A \
          B
          end{bmatrix} C =
          begin{bmatrix}
          AC \
          BC
          end{bmatrix}.
          $

        • $
          begin{bmatrix} A & B end{bmatrix} begin{bmatrix} C \ D end{bmatrix}
          = AC + BD.
          $


        (For each rule, we must assume that
        the matrices $A, B, C$, and $D$
        have compatible shapes.)



        Using these basic rules, we can easily derive any more complicated block matrix multiplication rule that we need. For example,



        begin{align}
        begin{bmatrix} A & B \ C & D end{bmatrix} begin{bmatrix} X \ Y end{bmatrix}
        &=
        begin{bmatrix}
        begin{bmatrix} A & B end{bmatrix} begin{bmatrix} X \ Y end{bmatrix} \
        begin{bmatrix} C & D end{bmatrix} begin{bmatrix} X \ Y end{bmatrix}
        end{bmatrix} \
        &=
        begin{bmatrix}
        AX + BY \
        CX + DY
        end{bmatrix}
        end{align}
        (assuming that the matrices $A, B, C, D, X$, and $Y$ have compatible shapes).






        share|cite|improve this answer











        $endgroup$



        This block matrix notation is very useful. The meaning of the notation is clear: for example, if
        $$
        A = begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} end{bmatrix}, quad
        B = begin{bmatrix} b_{11} & b_{12} \ b_{21} & b_{22} end{bmatrix}, quad
        C = begin{bmatrix} c_{11} & c_{12} \ c_{21} & c_{22} end{bmatrix}, quad
        D = begin{bmatrix} d_{11} & d_{12} \ d_{21} & d_{22} end{bmatrix}
        $$
        then $begin{bmatrix} A & B \ C & D end{bmatrix}$ denotes the $4 times 4$ matrix
        $$
        begin{bmatrix} A & B \ C & D end{bmatrix} =
        begin{bmatrix} a_{11} & a_{12} & b_{11} & b_{12} \
        a_{21} & a_{22} & b_{21} & b_{22} \
        c_{11} & c_{12} & d_{11} & d_{12} \
        c_{21} & c_{22} & d_{21} & d_{22}
        end{bmatrix}.
        $$
        It is straightforward to prove the following basic rules
        for matrix multiplication using block notation:





        • $A begin{bmatrix} B & C end{bmatrix} =
          begin{bmatrix} AB & AC end{bmatrix}$          .

        • $begin{bmatrix}
          A \
          B
          end{bmatrix} C =
          begin{bmatrix}
          AC \
          BC
          end{bmatrix}.
          $

        • $
          begin{bmatrix} A & B end{bmatrix} begin{bmatrix} C \ D end{bmatrix}
          = AC + BD.
          $


        (For each rule, we must assume that
        the matrices $A, B, C$, and $D$
        have compatible shapes.)



        Using these basic rules, we can easily derive any more complicated block matrix multiplication rule that we need. For example,



        begin{align}
        begin{bmatrix} A & B \ C & D end{bmatrix} begin{bmatrix} X \ Y end{bmatrix}
        &=
        begin{bmatrix}
        begin{bmatrix} A & B end{bmatrix} begin{bmatrix} X \ Y end{bmatrix} \
        begin{bmatrix} C & D end{bmatrix} begin{bmatrix} X \ Y end{bmatrix}
        end{bmatrix} \
        &=
        begin{bmatrix}
        AX + BY \
        CX + DY
        end{bmatrix}
        end{align}
        (assuming that the matrices $A, B, C, D, X$, and $Y$ have compatible shapes).







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Dec 18 '18 at 22:12

























        answered Dec 17 '18 at 11:39









        littleOlittleO

        30.1k647110




        30.1k647110






























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            Puebla de Zaragoza

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            Para otros usos de este término, véase Puebla (desambiguación). Heroica Puebla de Zaragoza Localidad De arriba a abajo de izquierda a derecha: Estrella de Puebla, Volcán Popocatépetl, Catedral de Puebla, Monumento a Ignacio Zaragoza, Estadio Cuauhtémoc, Angelópolis y Vista panorámica de la ciudad. Escudo Otros nombres: El relicario de América/ La ciudad de los Ángeles/ La Angelópolis Lema: Muy noble y muy leal ciudad de Puebla de los Ángeles Heroica Puebla de Zaragoza Localización de Heroica Puebla de Zaragoza en México Heroica Puebla de Zaragoza Localización de Heroica Puebla de Zaragoza en Puebla Mapa interactivo Coordenadas 19°03′05″N 98°13′04″O  /  19.051388888889, -98.217777777778 Coordenadas: 19°03′05″N 98°13′04″O  /  19.051388888889, -98.217777777778 Idioma oficial Español Entidad Localidad  • País   México  • Estado Puebla  • Municipio Puebla Presidente municipal Claudia Rivera Vivanc
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            Musa

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            Para el género de plantas, véase Musa (planta). Apolo y la Nueve Musas En la mitología griega, las musas (en griego antiguo μοῦσαι « mousai» ) son, según los escritores más antiguos, las divinidades inspiradoras de las Artes: cada una de ellas está relacionada con ramas artísticas y del conocimiento. Son hijas de Zeus y de Mnemósine, compañeras del séquito de Apolo, dios olímpico de la música y patrón de las bellas artes, quien tuvo romances con cada una de ellas, dejando descendientes. Bajaban a la tierra a susurrar ideas e inspirar a aquellos mortales que las invocaran. En la época más arcaica eran las ninfas inspiradoras de las fuentes, en las cuales eran adoradas. Finalmente, alrededor de los siglos VIII-VII a. C. [ 1 ] ​ prevaleció en todo el territorio de la Hélade la adoración de las nueve Musas, que son Calíope, Clío, Erato, Euterpe, Melpómene, Polimnia, Talía, Terpsícore y Urania. El culto a las musas era originalmente de Tracia y Beocia, y fueron de vital impo
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