diff --git a/chapters/vectors-matrix-ops/vectors-matrix-operations/media/25_4_Matrix_multiplication.png b/chapters/vectors-matrix-ops/vectors-matrix-operations/media/25_4_Matrix_multiplication.png new file mode 100644 index 0000000..45fbbbb Binary files /dev/null and b/chapters/vectors-matrix-ops/vectors-matrix-operations/media/25_4_Matrix_multiplication.png differ diff --git a/chapters/vectors-matrix-ops/vectors-matrix-operations/reading/README.md b/chapters/vectors-matrix-ops/vectors-matrix-operations/reading/README.md deleted file mode 100644 index 5d5a075..0000000 --- a/chapters/vectors-matrix-ops/vectors-matrix-operations/reading/README.md +++ /dev/null @@ -1 +0,0 @@ -# Vectors and Matrix Operations diff --git a/chapters/vectors-matrix-ops/vectors-matrix-operations/reading/read.md b/chapters/vectors-matrix-ops/vectors-matrix-operations/reading/read.md new file mode 100644 index 0000000..e3fcb3a --- /dev/null +++ b/chapters/vectors-matrix-ops/vectors-matrix-operations/reading/read.md @@ -0,0 +1,285 @@ +# Vectors and Matrix Operations + +## Numbers, Vectors, Matrices + +Recall that the set of real numbers is $\mathbb{R}$ and that a vector, $v \in \mathbb{R}^n$, is just an $n$-tuple of numbers. +Similarly, an $n \times m$ matrix is just a table of numbers, with $n$ rows and $m$ columns and we can write + +$$A_{mn} \in \mathbb{R}^{mn}$$ + +Note that a vector is normally considered equivalent to a $n\times 1$ matrix i.e. we view these as column vectors. + +### Examples + +:::info Example + +In R, a vector can be generated with: + +```text +> X <- 3:6 +> X +[1] 3 4 5 6 +``` + +A matrix can be generated in R as follows, + +```text +> matrix(X) + [,1] +[1,] 3 +[2,] 4 +[3,] 5 +[4,] 6 +``` + +::: + +:::note Note + +We note that R distinguishes between vectors and matrices. + +::: + +## Elementary Operations + +We can define multiplication of a real number $k$ and a vector $v=(v_1,\ldots,v_n)$ by $k\cdot v=(kv_1,\ldots,kv_n)$. +The sum of two vectors in $\mathbb{R}^n$, $v=(v_1,\ldots,v_n)$ and $u=(u_1,\ldots,u_n)$ is defined as the vector $v+u=(v_1+u_1,\ldots,v_n+u_n)$. +We can define multiplication of a number and a matrix and the sum of two matrices (of the same sizes) similarly. + +### Examples + +:::info Example + +```text +> A <- matrix(c(1,2,3,4), nr=2, nc=2) + +> A + [,1] [,2] +[1,] 1 3 +[2,] 2 4 + +> B <- matrix(c(1,0,2,1), nr=2, nc=2) + +> B + [,1] [,2] +[1,] 1 2 +[2,] 0 1 + +> A+B + [,1] [,2] +[1,] 2 5 +[2,] 2 5 +``` + +::: + +## The Tranpose of a Matrix + +In R, matrices may be constructed using the `matrix` function and the transpose of $A$, $A^\prime$, may be obtained in R by using the `t` function: + +```text +> A <- matrix(1:6, nrow=3) + +> t(A) + [,1] [,2] [,3] +[1,] 1 2 3 +[2,] 4 5 6 +``` + +### Details + +If $A$ is an $n \times m$ matrix with element $a_{ij}$ in row $i$ and column $j$, then $A^\prime$ or $A^T$ is the $m\times n$ matrix with element $a_{ij}$ in row $j$ and column $i$. + +### Examples + +:::info Example + +Consider a vector in R + +```text +> x <- 1:4 + +> x +[1] 1 2 3 4 + +> t(x) + [,1] [,2] [,3] [,4] +[1,] 1 2 3 4 + +> matrix(x) + [,1] +[1,] 1 +[2,] 2 +[3,] 3 +[4,] 4 +> t(matrix(x)) + [,1] [,2] [,3] [,4] +``` + +::: + +:::note Note + +The first solution gives a $1 \times n$ matrix and the second solution gives a $n \times 1$ matrix. + +::: + +## Matrix Multiplication + +Matrices $A$ and $B$ can be multiplied together if $A$ is an $n \times p$ matrix and $B$ is an $p\times m$ matrix. +The general element $c_{ij}$ of $n\times m$, $C=AB$, is found by pairing the $i^{th}$ row of $C$ with the $j^{th}$ column of $B$, and computing the sum of products of the paired terms. + +![Fig. 39](../media/25_4_Matrix_multiplication.png) + +### Details + +Matrices $A$ and $B$ can be multiplied together if $A$ is a $n\times p$ matrix and $B$ is a $p\times m$ matrix. +Given the general element $c_{ij}$ of $n \times m$ matrix, $C=AB$ is found by pairing the $i^{th}$ row of $C$ with the $j^{th}$ column of $B$, and computing the sum of products of the paired terms. + +### Examples + +:::info Example: Matrices in R + +```text +> A <- matrix(c(1,3,5,2,4,6),3,2) + +> A + [,1] [,2] +[1,] 1 2 +[2,] 3 4 +[3,] 5 6 + +> B <- matrix(c(1,1,2,3),2,2) + +> B + [,1] [,2] +[1,] 1 2 +[2,] 1 3 + +> A%*%B + [,1] [,2] +[1,] 3 8 +[2,] 7 18 +[3,] 11 28 +``` + +::: + +## More on Matrix Multiplication + +Let $A$, $B$, and $C$ be $m\times n$, $n\times l$, and $l\times p$ matrices, respectively. +Then we have + +$$(AB)C=A(BC)$$ + +In general, matrix multiplication is not commutative, that is $AB\neq BA$. + +We also have + +$$(AB)'=B'A'$$ + +In particular, $(Av)'(Av)=v'A'Av$, when $v$ is a $n\times1$ column vector + +More obvious are the rules + +1. $A+(B+C)=(A+B)+C$ + +1. $k(A+B)=kA+kB$ + +1. $A(B+C)=AB+AC$ + +where $k\in\mathbb{R}$ and when the dimensions of the matrices fit. + +## Linear Equations + +### Details + +General linear equations can be written in the form $Ax=b$. + +### Examples + +:::info Example + +The set of equations + +$$2x+3y=4$$ + +$$3x+y=2$$ + +can be written in matrix formulation as + +$$ +\begin{bmatrix} + 2 & 3 \\ + 3 & 1 +\end{bmatrix} +\begin{bmatrix} + x \\ + y +\end{bmatrix} = +\begin{bmatrix} + 4 \\ + 2 +\end{bmatrix} +$$ + +i.e. $A\underline{x} = \underline{b}$ for an appropriate choice of $A, \underline{x}$ and $\underline{b}$. + +::: + +## The Unit Matrix + +The $n\times n$ matrix + +$$ +I = + \left[ + \begin{array}{cccc} + 1 & 0 & \ldots & 0 \\ + 0 & 1 & 0 & \vdots \\ + \vdots & 0 & \dots & 0 \\ + 0 & \ldots & 0 & 1 + \end{array} + \right] +$$ + +is the identity matrix. +This is because if a matrix $A$ is $n\times n$ + +then $A I = A$ and $I A = A$ + +## The Inverse of a Matrix + +If $A$ is an $n \times n$ matrix and $B$ is a matrix such that + +$$BA = AB = I$$ + +then $B$ is said to be the inverse of $A$, written + +$$B = A ^{-1}$$ + +Note that if $A$ is an $n \times n$ matrix for which an inverse exists, then the equation $Ax = b$ can be solved and the solution is $x = A^{-1} b$. + +### Examples + +:::info Example + +If matrix $A$ is: + +$$ +\begin{bmatrix} + 2 & 3 \\ + 3 & 1 +\end{bmatrix} +$$ + +then $A ^{-1}$ is: + +$$ +\begin{bmatrix} + \displaystyle\frac{-1}{7} & \displaystyle\frac{3}{7} \\ + \displaystyle\frac{3}{7} & \displaystyle\frac{-2}{7} +\end{bmatrix} +$$ + +:::