Matrices: Linear Algebra

Algèbre linéaire

Matrices

• A matrix is a table of numbers with $m$ rows and $n$ columns.
• Elements are denoted as $a_{ij}$, where $i$ represents the row index and $j$ represents the column index.

Examples

• Example Matrix A = [[1, 2], [3, 4]].
• Example Matrix B = [[1, 2, 3], [4, 5, 6]].
• Example Matrix C = [[1], [2], [3]].

Operations

• Addition of matrices $A + B$ is performed element-wise: $[a_{ij} + b_{ij}]$, only if $A$ and $B$ have the same dimensions.
• Scalar multiplication $\lambda A$ is performed by multiplying each element of $A$ by the scalar $\lambda$: $[\lambda a_{ij}]$.
• Multiplication of matrices $(AB){ij}$ is computed as $\sum{k=1}^{n} a_{ik}b_{kj}$, requiring the number of columns in $A$ to equal the number of rows in $B$.

Properties

• Matrix addition is associative: $(A + B) + C = A + (B + C)$.
• Matrix addition is commutative: $A + B = B + A$.
• Scalar multiplication distributes over matrix addition: $\lambda(A + B) = \lambda A + \lambda B$.
• Scalar addition distributes over scalar multiplication of a matrix: $(\lambda + \mu)A = \lambda A + \mu A$.
• Matrix multiplication is associative: $A(BC) = (AB)C$.
• Matrix multiplication distributes over matrix addition: $A(B + C) = AB + AC$ and $(A + B)C = AC + BC$.

Transposition

• The transpose of a matrix $A$, denoted $A^T$, swaps rows and columns.
• The element at $(A^T){ij}$ equals $a{ji}$.
• $(A + B)^T = A^T + B^T$.
• $(\lambda A)^T = \lambda A^T$.
• $(AB)^T = B^T A^T$.

Special Matrices

• A square matrix has an equal number of rows and columns ($m = n$).
• A diagonal matrix has non-zero elements only on the main diagonal ($a_{ij} = 0$ if $i \neq j$).
• The identity matrix $I_n$ is a square matrix with 1s on the main diagonal and 0s elsewhere.
• A symmetric matrix is equal to its transpose: $A^T = A$.
• An antisymmetric matrix is the negative of its transpose: $A^T = -A$.

Systèmes linéaires

Definition

• A system of $m$ linear equations with $n$ unknowns is defined.

Matrix form

• The matrix is in the form $Ax = b$, where $A = [a_{ij}]$, $x = \begin{bmatrix} x_1 \ x_2 \ \vdots \ x_n \end{bmatrix}$, $b = \begin{bmatrix} b_1 \ b_2 \ \vdots \ b_m \end{bmatrix}$

Resolution

• Gauss elimination
• Back substitution

Solutions

• Unique
• Infinite
• None

3. Déterminants

Définition

• The determinant of a square matrix A, noted det(A) or |A|, is a scalar.

Calcul

• 2 x 2: det [[a, b], [c, d]] = ad - bc
• 3 x 3: Sarrus rule
• N x n: Expansion by row or column

Properties

• Det(A^T) = det(A)
• Det(AB) = det(A)det(B)
• If A has null row or column, then det(A) = 0
• If A has two identical rows or columns, then det(A) = 0
• If A' is obtained from A by swapping two rows or columns, then det(A') = -det(A)
• If A' is obtained from A by multiplying a row or column by a scalar $\lambda$, then det(A') = $\lambda$det(A)
• If A' is obtained from A by adding a row (or column) a multiple of another row (or column), then det(A') = det(A)

Inversibility

• A square matrix A is invertible if and only if det(A) $\neq$ 0. In this case, A^-1 = 1/det(A)adj(A), where adj(A) is the adjoint matrix A.

4. Espaces vectoriels

Definition

• A vector space is a set V with two operations:
• addition: u, v $\in$ V $\implies$ u + v $\in$ V
• Multiplication by a scaler: $\lambda$ $\in$ $\mathbb{R}$, u $\in$ V $\implies$ $\lambda$u $\in$ V
• satisfying certain properties (associativity, commutivity, existence of a neutral element an dan inverse)

Examples

• $\mathbb{R}^n$
• Polynomials
• Matrices

Sous-espaces vectoriels

• A subset W of V is a vector subspace if:
• 0 $\in$ W
• u, v $\in$ W $\implies u + $v $\in$ W
• $\lambda \in \mathbb{R}, u \in W \implies \lambda u \in W$

Base et dimension

• A base of V is a set of linearly independent vectors that generate V. The dimension of V is the number of vectors in a base.

5. Applications linéaires

Definition

• An application f: V -> W is linear if:
• f(u + v) = f(u) + f(v)
• f($\lambda$u) = $\lambda f(u)$

Examples

• Multiplication by a matrix
• Derivation
• Integration

Noyau et image

• Core: ker(f) = {u $\in$ V | f(u) = 0} The sum Image: im(f) = {f(u) | u $\in$ V}

Théorème du rang

• Dim(V) = dim(ker(f)) + dim(im(f))

6. Valeurs propres et vecteurs propres

Definition

• An eigenvector of A is a vector v such that Av = $\lambda$v, where $\lambda$ is an eigenvalue of A.

Calcul

• $\det(A - \lambda I) = 0$ (polynôme caractéristique)
• Solve for $\lambda$
• For each $\lambda$, resolve $(A - \lambda I)v = 0$ to find v

Diagonalisation

• A is diagonalizable of there exists an invertible matrix P such that $P^{-1}AP = D$, where D is a diagonal matrix. The columns of P are the eigenvectors of A.

Nature-Inspired Optimization Algorithm

Introduction

• Nature-Inspired Optimization Algorithms (NIOA) are computational methods of problem resolution inspired by nature
• Nature shows a rich repertoire of strategies to resolve complex optimization problems

What are Optimization Algorithms

• Optimization algorithms are computatational procedures designed to find the best possible solution for a given problem, among a possible list of solutions
• Optimization problems arise in engineering, economics, computer science, operations research

Why be nature-inspired

• Nature, through millions of evoluation, has refined efficient and robust solutions to complex problems
• NIOAs are inspired by natural phenomena, the evolution biological, the behavior of swarms and laws physics to develop algorithms optimization of the effectiveness

Types of Nature-Inspired Optimization Algorithms

Evolutionary Algorithms

• Algorithm inspired by the principles of natural selection and genetics, algorithms evolutionary instruments employ mechanisms such as reproduction, mutation, and recombination evolve to a population of candidate solutions over time.

Exemples

• Genetic Algorithm (GA)
• Evolution Strategies (ES)
• Genetic Programming (GP)

Algoritmos basados en el comportamiento de enjambres

• These algorithms imitate animal social collective behavior, like ants, bees and birds to solve problems optimization

Ejemplos

• Optimization by a colony of ants (Ant Colony Optimization, ACO)
• Optimization by a swarm of particles (Particle Swarm Optimization, PSO)
• Algoithm of the colony of artificial bees (Artificial Bee Colony Algorithm, ABC)

Other Alogrithms Inspired in Nature

• Simulated Annealing (SA): Simulated the process of metal cooling to find the global minimum of an objective function
• Harmony Search, HS: Inspired by the process of musical improvisation to find optimum solutions
• Optimization based in gravitational research (Gravitational Search Algorith, GSA): Based on the law of gravity and the mecanics newtonian

Applications of NIOA

• NIOA's demonstrated to be eficaces to resolve a wide rang of problems of optimization in differents fields. The common aplications are:
• Engineering: Structure designs, Route optimization, System control
• Computer Sciences: Machine learning, data mining, artificial vision
• Economy and finances: Optimization of portfolio. Market prediction
• Operative investigation: Task programming, logistics, Inventory management

Advantages and disadvantages

Advantages

• Flexibility: Can be applied to a wide rang of problems or optimization, including an objective unction not differentiable, not convex or bad
• Robustness: Are less sencible to optimal locals in comparation with algoriths of optimization of the traditional
• Paralelization: could it implemented easely in parallel architectures to accelerate the process optimization

Disadvantages

• Adjustment of parameters: Requires an ajustment careful with the parameters to get a perfomance optimum
• slow convergence: In some cases, they can verge slowly into the optimal solution
• Lack of guarantee optimal They don't garantizated to find the solution optimum global in all the cases

Conclusion

• The NIOA's offer an apporach promisor to resolve problems of optimization complex in diversity domains
• Its flexibility, robustnes and capability of paralelization convert it valued tool to broach challenges of optimization of the real world

Descriptive Statistics

Introduction

• Descriptive statistics is a branch of statistics that describes an abstract, and resumes data sets
• Used to organice, present and analyse data for easy interpretation

Definition

• Used to collect, orginace, abstract, present and analyse data with the objective of describing the principal caracteristics of a data set

Objectives

• The principal objectives are:
• Collect data: Get relevant information and representative of the population or the sample in study
• Organice data: Classificate and structure the data, so it's easy to analyse
• Abstract data: Calculate statistical measures that allow to represent the principal characteristics of the data
• Present data: Show the data in a way that is understandable, using charts, graphics, etc
• Analyse data: Identify patterns, trends and realtions in he data

Types of Statistical Variables

Qualitative Variable

• Qualitative variabes are representation or atributions not nummeric
• Qualitative nominal: Admits a order criteria.
• Ex: Civil state, favourite color
• Qualitative ordinal: There is a orden and hierarchy
• Ex: Level of studies, degree of satisfaction

Variable Cuantitative

• Quantitative Discrete: Toom a total values.
• Ex: Number of kids, number
• Cuantitative continous: can toom values inside a interval
• Ex: High, weight

Measures of Central Tendency

• Are values that represent the center in a data set

Arithmetic Media

• Is the promedy in a data set. Its calculates adding all the values end divide to the result by the number of data.
• $\qquad \bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}$

Media

• Is the value in the set
• the number of data is par, mean of the data

Fashion

• Is the data is more frequent

Measures of Dispensation

• The measure of despesation indicate the degree of variablility and dispensation is a media tendesy
• The measures used are standad and variarion coeffcient

Rang

• The rang is a differece between the value max y min in the set.
• $\qquad R = x_{max} - x_{min}$

Variance

• The sample variant is calculated by calculating the sums, devided and the mean
• $ s^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n-1}$
• Population Variants:
• $\qquad \sigma^2 = \frac{\sum_{i=1}^{N} (x_i - \mu)^2}{N}$

Standar Devision

• The deviation standar is the cuadratic coefficent in data in the originals the variance is calculated
• Sample estandar diasion
• $\qquad s = \sqrt{s^2} = \sqrt{\frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n-1}}$
• Sample estandar diasion
• $\qquad \sigma = \sqrt{\sigma^2} = \sqrt{\frac{\sum_{i=1}^{N} (x_i - \mu)^2}{N}}$

Variation Coeffcient

• The variation coeffcient is a measure that use standart devasion
• $\qquad CV = \frac{\sigma}{\mu} \cdot 100%$ (Población)
• $\qquad CV = \frac{s}{\bar{x}} \cdot 100%$ (Muestra)

Tablas de Frecuencia

Types of Freceency

• Frequency Absoluta ($f_i$): The number of times it appears in the data
• Relative frecuency ($h_i$): The porpotion of the value it apears. devide the number $\qquad h_i = \frac{f_i}{n}$
• acumulado
• relativa acumulada

Chapter 2 - Vector Spaces

2.1 Definition 2.1.1 - Internal Composition Law

• Let E be a non-empty set.
• An internal composition law (LCI) on E is an application from $E \times E$ to $E$, mapping $(x, y)$ to $x * y$.

2.1.2 (Group)

• A group is a set G with an internal composition law *, satisfying these properties:
  • Associativity: For all x, y, z in G, $(x * y) * z = x * (y * z)$.
  • Existence of a neutral element: There exists e in G such that for all x in G, $x * e = e * x = x$.
  • Existence of an inverse: For all x in G, there exists $x^{-1}$ in G such that $x * x^{-1} = x^{-1} * x = e$.

2.1.3 (Commutative Group)

• A group (G, *) is commutative or abelian if the internal composition law * is commutative, meaning that for all x, y in G: $x * y = y * x$.

2.1.4 (External Compostion Law)

• Let K be a field and E a non-empty set.
• External composition Law

2.1.5 (Vector Space)

• Let K be a commutative field and E a non-empty set. E is a vector space over K, if E has an internal composition law + and an external composition with operators in K, denoted by ., satisfying the following properties:
  • (E, +) is a commutative group.
  • For all $\lambda, \mu \in K$, for all $x \in E$: $(\lambda + \mu). x = \lambda. x + \mu. x$.
  • For all $\lambda \in K$, for all $x, y \in E$: $\lambda. (x + y) = \lambda. x + \lambda. y$.
  • For all $\lambda, \mu \in K$, for all $x \in E$: $(\lambda \mu). x = \lambda. (\mu. x)$.
  • For all $x \in E: 1. x = x$, where 1 is the neutral element for multiplication in K.
• Elements of E are called vectors, and elements of K are called scalars.

Physics

Mechanics

• Rectilinear motion.
• Motion in a plane.
• Newton's laws of motion.
• Work, energy, and power.
• System of particles.
• Rotational motion.
• Gravitation.
• Mechanical properties of solids.
• Mechanical properties of fluids.
• Thermal properties of matter.
• Thermodynamics.
• Kinetic theory.

Oscillations and Waves

• Oscillations.
• Waves.

Electrostatics

• Electric charges and fields.
• Electrostatic potential and capacitance.

Current Electricity

• Current electricity.

Magnetic Effects of Current and Magnetism

• Moving charges and magnetism.
• Magnetism and matter.

Electromagnetic Induction and Alternating Currents

• Electromagnetic induction.
• Alternating currents.

Electromagnetic Waves

• Electromagnetic waves.

Optics

• Ray optics and optical instruments.
• Wave optics.

Modern Physics

• Dual nature of radiation and matter.
• Atoms.
• Nuclei.
• Semiconductor electronics involving materials, devices, and simple circuits.