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Let a b and c all be matrices. Is a(b dot c) the same as (ab dot ac)? Or what is the right way to expand this? Note that it's not cross product but standard matrix multiplication (or left product)

Lucy Gu
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  • So what are the two operations here. One is normal matrix multiplication, and the other is? – Rushabh Mehta Jan 24 '21 at 21:41
  • Why don't you try a few examples to test your questions. Also note that there are restrictions on the size of the matrices. – MasB Jan 24 '21 at 21:42
  • Is the "dot" supposed to be addition? Because why would you think $A(B\cdot C) = AB \cdot AC$? However, it is true that $A(B+C)=AB+AC$ because matrices are distributive under addition – Eminem Jan 24 '21 at 21:42
  • Does "ab" mean "a dot b"? –  Jan 24 '21 at 21:45
  • No. $A\cdot (B\cdot C) \ne (A\cdot B)\cdot (A\cdot C)$. You seem to be asking if the operation you are using to combine $A$ with $B$ and/or $C$ will distribute across the operation used to combine $B$ with $C$. This seems parallel with multiplication distributing across addition: $x(y + z) = xy + xz$. But as the operations are both dot product, dot product does not distribute across itself. (almost no operations distribute across themselves and must pairs of operations don't distribute). In general $a\dagger(b\circ c)\ne a\dagger b \circ a\dagger c$ unless you have a good reason. – fleablood Jan 24 '21 at 22:01
  • Notice if $B= C=I=$ the identity matrice then $A(B\cdot C) = A(I \cdot I) =A\cdot I = A$. But $(A\cdot B)(A\cdot C) = (A\cdot I)(A\cdot I) = A\cdot A = A^2$. ANd $A \ne A^2$ in general. So, no, this can't be true. – fleablood Jan 24 '21 at 22:04
  • I'm not sure what you mean by the right way to expand $A(BC)$. As that figuring out what $X = BC$ is and then doing $AX$ the right way to express it IS as $A(BC)$. and there is nothing further to do. As matrix multiplication is associative we do know that $A(BC) = (AB)C$ and that doing $X = BC$ and figuring out what $AX$ is is the same thing as doing $Y=AB$ and figuring out what $YC$ is, we do not need to write the parantehsis. Writing, $ABC$ to mean $(AB)C$ which is equal to $A(BC)$ will be understood. – fleablood Jan 24 '21 at 22:12
  • Sorry about any confusion. I don't know how to type "dot" with my computer so the question is asking: b dot c is dot product. Then, this dot product (which should be a scalar?) is multiplied by a matrix a by matrix multiplication. Then I wanted to ask if it is same as ab dot ac, where ab is matrix multiplication between a and b, and ac is matrix multiplication between a and c – Lucy Gu Jan 24 '21 at 22:38
  • Sorry about any confusion. I don't know how to type "dot" with my computer so the question is asking: b dot c is dot product. Then, this dot product (which should be a scalar?) is multiplied by a matrix a by matrix multiplication. Then I wanted to ask if it is same as ab dot ac, where ab is matrix multiplication between a and b, and ac is matrix multiplication between a and c – Lucy Gu Jan 24 '21 at 22:38
  • Sorry about any confusion. I don't know how to type "dot" with my computer so the question is asking: b dot c is dot product. Then, this dot product (which should be a scalar?) is multiplied by a matrix a by matrix multiplication. Then I wanted to ask if it is same as ab dot ac, where ab is matrix multiplication between a and b, and ac is matrix multiplication between a and c – Lucy Gu Jan 24 '21 at 22:39

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