This post is part of my Game Math Series.
Definition of Matrix Multiplication
Let 
and 
be a 
-by-
matrix and an -by-
matrix, respectively. Let 
denote the element in the matrix 
on row 
and column 
(zero-based indexing), “element (, )” for short.
From the definition of matrix multiplication, we know that:
![\[ (AB)_{ij} = \sum\limits_{k = 0}^{m - 1} {A_{ik} B_{kj}} \]](https://allenchou.net/wp-content/ql-cache/quicklatex.com-95dbdea036a6e37f283e6e1d3e7600e7_l3.png "Rendered by QuickLaTeX.com")
Essentially, element (, ) of the matrix product 
is the dot product of -th row from and -th column from 
.
We can compute matrix multiplication by performing one dot product per element in the resulting matrix product . However, there are two alternative ways to look at the matrix multiplication operation that can sometimes make your life easier. I learned these neat tips from the “Linear Algebra and Its Applications” course when I was studying at National Taiwan University back in Taiwan.
Linear Combination of Columns
The -th column of the matrix product is a linear combination of columns from the left matrix , where the coefficients of linear combination are from the -th column of the right matrix .
Let’s look at a product of two 3-by-3 matrices as an example:
![\[ A = {\left[ {\begin{array}{ccc} A_{00} & A_{01} & A_{02} \\ A_{10} & A_{11} & A_{12} \\ A_{20} & A_{21} & A_{22} \\ \end{array} } \right]} = {\left[ {\begin{array}{ccc} C_A^0 & C_A^1 & C_A^2 \\ \end{array} } \right]} , \]](https://allenchou.net/wp-content/ql-cache/quicklatex.com-60b8115d2131d1d92ff05716c4fb8ef6_l3.png "Rendered by QuickLaTeX.com")
![\[ B = {\left[ {\begin{array}{ccc} B_{00} & B_{01} & B_{02} \\ B_{10} & B_{11} & B_{12} \\ B_{20} & B_{21} & B_{22} \\ \end{array} } \right]}, \]](https://allenchou.net/wp-content/ql-cache/quicklatex.com-af325715559cb29c273e9a5964dbea49_l3.png "Rendered by QuickLaTeX.com")
where 
denotes the -th column of matrix .
Then the -th column of the matrix product is:
![\[ C_{AB}^i = B_{0i} C_A^0} + B_{1i} C_A^1 + B_{2i} C_A^2 \]](https://allenchou.net/wp-content/ql-cache/quicklatex.com-ec9bcd772b71411e9a1b28c43eb488f8_l3.png "Rendered by QuickLaTeX.com")
Now let’s look at an example with real numbers:
![\[ A = {\left[ {\begin{array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \\ \end{array} } \right]} , B = {\left[ {\begin{array}{ccc} 1 & -1 & 0 \\ 2 & 1 & -2 \\ 0 & 2 & 3 \\ \end{array} } \right]} \]](https://allenchou.net/wp-content/ql-cache/quicklatex.com-3345a183f4c2861eaea0819c3cb0004b_l3.png "Rendered by QuickLaTeX.com")
The columns of are ![{\left[ {\begin{array}{c} 1 \\ 4 \\ 7 \end{array} } \right]}](https://allenchou.net/wp-content/ql-cache/quicklatex.com-da366fc447e9d93d8d1ea73bd5a37974_l3.png "Rendered by QuickLaTeX.com")
, ![{\left[ {\begin{array}{c} 2 \\ 5 \\ 8 \end{array} } \right]}](https://allenchou.net/wp-content/ql-cache/quicklatex.com-3cd371798bf02ef5fd9d136c5fc42a23_l3.png "Rendered by QuickLaTeX.com")
, and ![{\left[ {\begin{array}{c} 3 \\ 6 \\ 9 \end{array} } \right]}](https://allenchou.net/wp-content/ql-cache/quicklatex.com-057aa359a8226599a132d21227c9b68f_l3.png "Rendered by QuickLaTeX.com")
. The first column of is ![{\left[ {\begin{array}{c} 1 \\ 2 \\ 0 \end{array} } \right]}](https://allenchou.net/wp-content/ql-cache/quicklatex.com-13903bd1b7429254f55160db9162c335_l3.png "Rendered by QuickLaTeX.com")
, so the first column of the matrix product is:
![\[ C_{AB}^0 = 1 {\left[ {\begin{array}{c} 1 \\ 4 \\ 7 \end{array} } \right]} + 2 {\left[ {\begin{array}{c} 2 \\ 5 \\ 8 \end{array} } \right]} + 0 {\left[ {\begin{array}{c} 3 \\ 6 \\ 9 \end{array} } \right]} = {\left[ {\begin{array}{c} 5 \\ 14 \\ 23 \end{array} } \right]} \]](https://allenchou.net/wp-content/ql-cache/quicklatex.com-4a1a2d64aee11af71b955a0030093a78_l3.png "Rendered by QuickLaTeX.com")
Similarly, the second and third columns of the matrix product are:
![\[ C_{AB}^1 = -1 {\left[ {\begin{array}{c} 1 \\ 4 \\ 7 \end{array} } \right]} + 1 {\left[ {\begin{array}{c} 2 \\ 5 \\ 8 \end{array} } \right]} + 2 {\left[ {\begin{array}{c} 3 \\ 6 \\ 9 \end{array} } \right]} = {\left[ {\begin{array}{c} 7 \\ 13 \\ 19 \end{array} } \right]} \]](https://allenchou.net/wp-content/ql-cache/quicklatex.com-85d5eeb68fa32841d7449fb1b66da2ea_l3.png "Rendered by QuickLaTeX.com")
![\[ C_{AB}^2 = 0 {\left[ {\begin{array}{c} 1 \\ 4 \\ 7 \end{array} } \right]} - 2 {\left[ {\begin{array}{c} 2 \\ 5 \\ 8 \end{array} } \right]} + 3 {\left[ {\begin{array}{c} 3 \\ 6 \\ 9 \end{array} } \right]} = {\left[ {\begin{array}{c} 5 \\ 8 \\ 11 \end{array} } \right]} \]](https://allenchou.net/wp-content/ql-cache/quicklatex.com-5392714d4015d64ba949a309316a383d_l3.png "Rendered by QuickLaTeX.com")
So the matrix product is:
![\[ AB = {\left[ {\begin{array}{ccc} 5 & 7 & 5 \\ 14 & 13 & 8 \\ 23 & 19 & 11 \\ \end{array} } \right]} \]](https://allenchou.net/wp-content/ql-cache/quicklatex.com-56c161bf9944709d5526d8fd0ca8f569_l3.png "Rendered by QuickLaTeX.com")
Linear Combination of Rows
Alternatively, we can view matrix multiplication from a row perspective.
The -th row of the matrix product is a linear combination of rows from the right matrix , where the coefficients of linear combination are from the -th row of the left matrix .
Let 
denote the -th row of matrix .
I will skip the symbolic notations and jump right back into the previous example we used:
The rows of are ![{\left[ {\begin{array}{ccc} 1 & -1 & 0 \end{array} } \right]}](https://allenchou.net/wp-content/ql-cache/quicklatex.com-bbdc0d1e02d0fe1fad7614252e657f14_l3.png "Rendered by QuickLaTeX.com")
, ![{\left[ {\begin{array}{ccc} 2 & 1 & -2 \end{array} } \right]}](https://allenchou.net/wp-content/ql-cache/quicklatex.com-de360793016916b228a7f9fc43e62dd8_l3.png "Rendered by QuickLaTeX.com")
, and ![{\left[ {\begin{array}{ccc} 0 & 2 & 3 \end{array} } \right]}](https://allenchou.net/wp-content/ql-cache/quicklatex.com-5b7e57d676de544a69c64a4fb85f14db_l3.png "Rendered by QuickLaTeX.com")
. The first rowof is ![{\left[ {\begin{array}{ccc} 1 & 2 & 3 \end{array} } \right]}](https://allenchou.net/wp-content/ql-cache/quicklatex.com-fabba2a004dc95c9bb0deaf0f863ddb6_l3.png "Rendered by QuickLaTeX.com")
, so the first column of the matrix product is:
![\[ R_{AB}^0 = 1 {\left[ {\begin{array}{ccc} 1 & -1 & 0 \end{array} } \right]} + 2 {\left[ {\begin{array}{ccc} 2 & 1 & -2 \end{array} } \right]} + 3 {\left[ {\begin{array}{ccc} 0 & 2 & 3 \end{array} } \right]} = {\left[ {\begin{array}{ccc} 5 & 7 & 5 \end{array} } \right]} \]](https://allenchou.net/wp-content/ql-cache/quicklatex.com-1672e8f10556c78e1baf540cb337d72d_l3.png "Rendered by QuickLaTeX.com")
Similarly, the second and third rows of the matrix product are:
![\[ R_{AB}^1 = 4 {\left[ {\begin{array}{ccc} 1 & -1 & 0 \end{array} } \right]} + 5 {\left[ {\begin{array}{ccc} 2 & 1 & -2 \end{array} } \right]} + 6 {\left[ {\begin{array}{ccc} 0 & 2 & 3 \end{array} } \right]} = {\left[ {\begin{array}{ccc} 14 & 13 & 8 \end{array} } \right]} \]](https://allenchou.net/wp-content/ql-cache/quicklatex.com-479a16c94bbdfb8eeceed8d47df9fc43_l3.png "Rendered by QuickLaTeX.com")
![\[ R_{AB}^2 = 7 {\left[ {\begin{array}{ccc} 1 & -1 & 0 \end{array} } \right]} + 8 {\left[ {\begin{array}{ccc} 2 & 1 & -2 \end{array} } \right]} + 9 {\left[ {\begin{array}{ccc} 0 & 2 & 3 \end{array} } \right]} = {\left[ {\begin{array}{ccc} 23 & 19 & 11 \end{array} } \right]} \]](https://allenchou.net/wp-content/ql-cache/quicklatex.com-6aff5401d47d3b474f75853bbfa1d184_l3.png "Rendered by QuickLaTeX.com")
So the matrix product is:
![\[ AB = {\left[ {\begin{array}{ccc} 5 & 7 & 5 \\ 14 & 13 & 8 \\ 23 & 19 & 11 \\ \end{array} } \right]} , \]](https://allenchou.net/wp-content/ql-cache/quicklatex.com-1839cf40e35b595cab7cdf2521dfa80e_l3.png "Rendered by QuickLaTeX.com")
same as what we got before.
End of Alternate Views on Matrix Multiplication
Now that you understand the two alternative views of matrix multiplication, you are well equipped to make your life easier when dealing with various matrix operations. For instance, I will show how to quickly eyeball the inverse of small matrices using this technique in another post.
