Matrix

13 Jan 2018 on Math, Linear algebra

1. Scala

$x \in \mathbf{R}$

2. vector

$x = \begin{bmatrix}x_{1} \\x_{2} \\\vdots \\x_{N} \\\end{bmatrix},\;\;\;\;x \in \mathbf{R}^N$

3. Matrix

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- Determinant

2X2

${\begin{aligned}|A|={\begin{vmatrix}a&b\\c&d\end{vmatrix}}=ad-bc.\end{aligned}}$

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3X3

${\begin{aligned}|A|={\begin{vmatrix}a&b&c\\d&e&f\\g&h&i\end{vmatrix}}&=a\,{\begin{vmatrix}e&f\\h&i\end{vmatrix}}-b\,{\begin{vmatrix}d&f\\g&i\end{vmatrix}}+c\,{\begin{vmatrix}d&e\\g&h\end{vmatrix}}\\&=aei+bfg+cdh-ceg-bdi-afh.\end{aligned}}$

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- Inverse Matrix

$A^{-1}={\frac {1}{\det(A)}}\operatorname {adj} (A)$

4. Vector Algebra

- Length

$\left\|\mathbf {a} \right\|={\sqrt ^{2}+{a_{2}}^{2}+{a_{3}}^{2}}}$

- Unit Vector

$\mathbf {\hat {a}} ={\frac {\mathbf {a} }{\left\|\mathbf {a} \right\|}}={\frac {a_{1}}{\left\|\mathbf {a} \right\|}}\mathbf {e} _{1}+{\frac {a_{2}}{\left\|\mathbf {a} \right\|}}\mathbf {e} _{2}+{\frac {a_{3}}{\left\|\mathbf {a} \right\|}}\mathbf {e} _{3}$

- Addition

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- Subtraction

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- Scalar Multiplication

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- Dot Product

$\mathbf {a} \cdot \mathbf {b} =\left\|\mathbf {a} \right\|\left\|\mathbf {b} \right\|\cos \theta$

$\mathbf {a} \cdot \mathbf {b} =\sum _{i=1}^{n}a_{i}b_{i}=a_{1}b_{1}+a_{2}b_{2}+\cdots +a_{n}b_{n}$

- Cross Product

$\mathbf {a} \times \mathbf {b} =\left\|\mathbf {a} \right\|\left\|\mathbf {b} \right\|\sin(\theta )\,\mathbf {n}$

${\mathbf {a} }\times {\mathbf {b} }=(a_{2}b_{3}-a_{3}b_{2}){\mathbf {e} }_{1}+(a_{3}b_{1}-a_{1}b_{3}){\mathbf {e} }_{2}+(a_{1}b_{2}-a_{2}b_{1}){\mathbf {e} }_{3}.$

Reference

Vector algebra - wikipedia

Matrix (math) - wikipedia

Determinant - wikipedia

Euclidean vector - wikipedia