Applied Mathematics/The Basics

${\displaystyle \mathbf {A} ={\begin{bmatrix}a_{11}&a_{12}&a_{12}&a_{14}\\a_{21}&a_{22}&a_{23}&a_{24}\\a_{31}&a_{32}&a_{33}&a_{34}\\\end{bmatrix}}.}$

A matrix is composed of a rectangular array of numbers arranged in rows and columns. The horizontal lines are called rows and the vertical lines are called columns. The individual items in a matrix are called elements. The element in the i-th row and the j-th column of a matrix is referred to as the i,j, (i,j), or (i,j)th element of the matrix. To specify the size of a matrix, a matrix with m rows and n columns is called an m-by-n matrix, and m and n are called its dimensions.

Operation	Definition	Example
Addition	The sum A+B of two m-by-n matrices A and B is calculated entrywise: (A + B)i,j = Ai,j + Bi,j, where 1 ≤ i ≤ m and 1 ≤ j ≤ n.	${\displaystyle {\begin{bmatrix}1&3&1\\1&0&0\end{bmatrix}}+{\begin{bmatrix}0&0&5\\7&5&0\end{bmatrix}}={\begin{bmatrix}1+0&3+0&1+5\\1+7&0+5&0+0\end{bmatrix}}={\begin{bmatrix}1&3&6\\8&5&0\end{bmatrix}}}$
Scalar multiplication	The scalar multiplication cA of a matrix A and a number c (also called a scalar in the parlance of abstract algebra) is given by multiplying every entry of A by c: (cA)i,j = c · Ai,j.	${\displaystyle 2\cdot {\begin{bmatrix}1&8&-3\\4&-2&5\end{bmatrix}}={\begin{bmatrix}2\cdot 1&2\cdot 8&2\cdot -3\\2\cdot 4&2\cdot -2&2\cdot 5\end{bmatrix}}={\begin{bmatrix}2&16&-6\\8&-4&10\end{bmatrix}}}$
Transpose	The transpose of an m-by-n matrix A is the n-by-m matrix AT (also denoted Atr or tA) formed by turning rows into columns and vice versa: (AT)i,j = Aj,i.	${\displaystyle {\begin{bmatrix}1&2&3\\0&-6&7\end{bmatrix}}^{\mathrm {T} }={\begin{bmatrix}1&0\\2&-6\\3&7\end{bmatrix}}}$

(1) ${\displaystyle {\begin{bmatrix}5&7&3\\1&2&9\end{bmatrix}}+{\begin{bmatrix}4&0&5\\8&3&0\end{bmatrix}}=}$
(2) ${\displaystyle 4{\begin{bmatrix}-1&0&-5\\7&9&-6\end{bmatrix}}=}$
(3) ${\displaystyle {\begin{bmatrix}-2&5&7\\0&0&9\end{bmatrix}}^{\mathrm {T} }=}$

Multiplication of two matrices is defined only if the number of columns of the left matrix is the same as the number of rows of the right matrix. If A is an m-by-n matrix and B is an n-by-p matrix, then their matrix product AB is the m-by-p matrix whose entries are given by dot product of the corresponding row of A and the corresponding column of B^[2]

${\displaystyle [\mathbf {AB} ]_{i,j}=A_{i,1}B_{1,j}+A_{i,2}B_{2,j}+\cdots +A_{i,n}B_{n,j}=\sum _{r=1}^{n}A_{i,r}B_{r,j}}$^[3]

${\displaystyle {\begin{bmatrix}-2&0\\3&2\end{bmatrix}}{\begin{bmatrix}1&2\\3&-1\end{bmatrix}}}$

${\displaystyle ={\begin{bmatrix}-2+0&-4+0\\3+6&6+(-2)\end{bmatrix}}}$

${\displaystyle ={\begin{bmatrix}-2&-4\\9&4\end{bmatrix}}}$

(1) ${\displaystyle {\begin{bmatrix}1&0\\2&2\end{bmatrix}}{\begin{bmatrix}4\\2\end{bmatrix}}=}$

(2) ${\displaystyle {\begin{bmatrix}1&2\\2&3\end{bmatrix}}{\begin{bmatrix}2&3\\1&4\end{bmatrix}}=}$

A row vector is a 1 × m matrix, while a column vector is a m × 1 matrix.

Suppose A is row vector and B is column vector, then the dot product is defined as follows;

${\displaystyle A\cdot B=|A||B|cos\theta }$

or

${\displaystyle \mathbf {A} \cdot \mathbf {B} ={\begin{pmatrix}a_{1}&a_{2}&\cdots &a_{n}\end{pmatrix}}{\begin{pmatrix}b_{1}\\b_{2}\\\vdots \\b_{n}\end{pmatrix}}=a_{1}b_{1}+a_{2}b_{2}+\cdots +a_{n}b_{n}=\sum _{i=1}^{n}a_{i}b_{i}}$

Suppose ${\displaystyle \mathbf {A} ={\begin{pmatrix}a_{1}&a_{2}&a_{3}\end{pmatrix}}}$ and ${\displaystyle \mathbf {B} ={\begin{pmatrix}b_{1}\\b_{2}\\b_{3}\end{pmatrix}}}$ The dot product is

${\displaystyle \mathbf {A} \cdot \mathbf {B} ={\begin{pmatrix}a_{1}&a_{2}&a_{3}\end{pmatrix}}{\begin{pmatrix}b_{1}\\b_{2}\\b_{3}\end{pmatrix}}=a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}}$

Suppose ${\displaystyle \mathbf {A} ={\begin{pmatrix}2\\1\\3\end{pmatrix}}}$ and ${\displaystyle \mathbf {B} ={\begin{pmatrix}7\\5\\4\end{pmatrix}}}$

${\displaystyle \mathbf {A} \cdot \mathbf {B} ={\begin{pmatrix}2&1&3\end{pmatrix}}{\begin{pmatrix}7\\5\\4\end{pmatrix}}}$

${\displaystyle =2\cdot 7+1\cdot 5+3\cdot 4}$
${\displaystyle =14+5+12}$

${\displaystyle =31}$

(1) ${\displaystyle \mathbf {A} ={\begin{pmatrix}3\\2\\5\end{pmatrix}}}$ and ${\displaystyle \mathbf {B} ={\begin{pmatrix}1\\4\\3\end{pmatrix}}}$

${\displaystyle \mathbf {A} \cdot \mathbf {B} =}$

(2) ${\displaystyle \mathbf {A} ={\begin{pmatrix}1\\0\\3\end{pmatrix}}}$ and ${\displaystyle \mathbf {B} ={\begin{pmatrix}6\\9\\2\end{pmatrix}}}$

${\displaystyle \mathbf {A} \cdot \mathbf {B} =}$

Cross product is defined as follows:

${\displaystyle A\times B=|A||B|sin\theta }$

Or, using detriment,

${\displaystyle \mathbf {A\times B} ={\begin{vmatrix}e_{x}&e_{y}&e_{z}\\a_{x}&a_{y}&a_{z}\\b_{x}&b_{y}&b_{z}\\\end{vmatrix}}=(a_{y}b_{z}-a_{z}b_{y},a_{z}b_{x}-a_{x}b_{z},a_{x}b_{y}-a_{y}b_{x})}$

where ${\displaystyle e}$ is unit vector.