diff --git a/docs/ce1/math115.md b/docs/ce1/math115.md index e9d3951..2b942f3 100644 --- a/docs/ce1/math115.md +++ b/docs/ce1/math115.md @@ -416,6 +416,34 @@ Properties of transposed matrices: - $(A^T)^T = A$ - $(A+B)^T=A^T+B^T$ +### Matrix multiplication + +In the two matrices $A\in M_{m\times n}(\mathbb R)$ and $B\in M_{n\times k}(\mathbb R)$, where $B=[\vec b_1, ..., \vec b_n]$ are columns: + +$$AB=[A\vec b_1, ..., A\vec b_n]$$ + +where the resultant matrix is of size $m\times k$. + +Extra columns in $B$ are ignored. + +Alternatively, where $r_i$ is each row in $A$: + +$$ +AB = \begin{bmatrix} +\vec r_1\bullet \vec b_1 & ... & \vec r_1\bullet b_k \\ +... & ... & ... \\ +\vec r_m\bullet b_1 & ... & \vec r_m\bullet \vec b_k +\end{bmatrix} +$$ + +Properties of matrix multiplication, where $x\in \mathbb R$: + +- $(AB)^T = B^T + A^T$ +- $A(BC) = A(BC)$ +- $AB \neq BA$ +- $(B+C)A = BA+ BC$ +- $(xA)B = x(AB) = A(xB)$ + ### Matrix-vector product