Matrices: Introduction and Operations

Matrices Introduction

• A matrix is a rectangular array of numbers or symbols in rows and columns.
• Examples include 2x3, 2x2, and 3x1 matrices denoted as A, B, and C respectively.

Basic Matrix Operations

• Matrices can be added or subtracted if they have the same dimensions, by adding or subtracting corresponding elements.
• Scalar multiplication involves multiplying each element of the matrix by the scalar.
• For matrices A (m x n) and B (n x p), matrix multiplication results in an m x p matrix, with elements computed as (AB)${ij}$ = $\sum{k=1}^{n} a_{ik}b_{kj}$.

Special Matrices

• An identity matrix is a square matrix with 1s on the main diagonal and 0s elsewhere (e.g., a 2x2 identity matrix).
• The transpose of a matrix A ($A^T$) is obtained by interchanging its rows and columns.
• The inverse of a square matrix A ($A^{-1}$) satisfies $AA^{-1}$ = $A^{-1}A$ = I, where I is the identity matrix.

Determinant

• A determinant is a scalar value computed from the elements of a square matrix.
• For a 2x2 matrix A = $\begin{bmatrix} a & b \ c & d \end{bmatrix}$, det(A) = ad - bc

Applications of Matrices

• Matrices are used in solving systems of linear equations.
• Matrices are used in linear transformations.
• Matrices are used in computer graphics.
• Matrices are used in data analysis.

The Simplex Method Overview

• The simplex method is an algebraic approach used to solve linear programming problems.

Setting Up the Problem

• Inequality constraints are converted to equality constraints by introducing slack variables ($s_1, s_2, s_3 \geq 0$).
• The objective function is rewritten to equal zero ($Z - 3x_1 - 5x_2 = 0$).
• Basic variables are $s_1, s_2, s_3, Z$, and nonbasic variables are $x_1, x_2$.
• Initial solution obtained is by setting nonbasic variables to zero: $(x_1, x_2, s_1, s_2, s_3, Z) = (0, 0, 4, 12, 18, 0)$.

Simplex Tableau

• The initial simplex tableau organizes the coefficients of the variables and slack variables, along with the right-hand side (RHS) values.

Performing Row Operations

• The entering variable is selected based on the most negative entry in the bottom row of the tableau.
• The departing variable is determined by the smallest nonnegative ratio.
• Row operations transform the tableau, making other entries in the entering variable column zero and update the solution.
• In the provided example, optimal solution is $x_1 = 2, x_2 = 6, Z = 36$.

Problem objectives for water production:

• Maximize total distilled water production.
• Ensure production of each unit does not exceed its maximum capacity.

Variables:

• $x_1$ represents quantity of water distilled by Unit 1.
• $x_2$ represents quantity of water distilled by Unit 2.

Constraints:

• The minimum demand constraint ensures that total production meets at least 8 tons, described as $x_1 + x_2 \geq 8$.
• Unit 1's production cannot exceed 5 tons, represented as $x_1 \leq 5$.
• Unit 2's production cannot exceed 7 tons, represented as $x_2 \leq 7$.
• The non-negativity constraint ensures production quantities are not negative, $x_1 \geq 0, x_2 \geq 0$.

Objective Function:

• Maximize total water production: $max \ z = x_1 + x_2$.

Linear Algebra and Analytic Geometry Overview

• Focus on systems of linear equations.

Linear Equations (1.1)

• Def: Linear Equation with n variables ($x_1, x_2,..., x_n$) : $a_1x_1 + a_2x_2 +... + a_nx_n = b$, where $a_1, a_2,..., a_n, b$ are real constants.
• Def: A System of $m$ Linear Equations with $n$ variables is a set of $m$ linear equations equations system, each one with $n$ variables. A general form is given as such for $1 \leq i \leq m$ and $1 \leq j \leq n$: $$\begin{cases} a_{11}x_1 + a_{12}x_2 +... + a_{1n}x_n = b_1 \ a_{21}x_1 + a_{22}x_2 +... + a_{2n}x_n = b_2 \... \ a_{m1}x_1 + a_{m2}x_2 +... + a_{mn}x_n = b_m \end{cases}$$

Types of Linear Systems (1.2)

• A linear system is compatible if it has at least one solution; incompatible indicates no solution.
• Compatible Determined: a unique solution
• Compatible Indetermined: infinite solutions
• Homogeneous systems have all constant terms equal to zero. $$\begin{cases} a_{11}x_1 + a_{12}x_2 +... + a_{1n}x_n = 0 \ a_{21}x_1 + a_{22}x_2 +... + a_{2n}x_n = 0 \... \ a_{m1}x_1 + a_{m2}x_2 +... + a_{mn}x_n = 0 \end{cases}$$
• Homogeneous systems are always compatible as they have a trivial solution where $x_1 = x_2 =... = x_n = 0$

Triboelectric Series

• Materials in the list become positively charged when rubbed with materials above it and negatively charged when rubbed with materials below it.
• Lists materials from Air to Teflon

Charging Methods

• Friction involves the transfer of electrons between objects.
• Conduction happens through direct contact, allowing electrons to move.
• Induction rearranges charges in an object without direct contact.

Coulomb's Law

• Describes the electrostatic force (F) between two charges: $F = k \frac{q_1 q_2}{r^2}$
• $k \approx 9 \times 10^9 \frac{N \cdot m^2}{C^2}$, $q_1 \text{ and } q_2$ are charge magnitudes and $r$ is the distance.

Charge of Fundamental Particles

• Proton: $+1.6 \times 10^{-19}$ C
• Electron: $-1.6 \times 10^{-19}$ C
• Neutron: 0 C

Electric Fields

• Defined as $E = \frac{F}{q}$, where E is the electric field, F is the electrostatic force, and q is the charge.
• Electric field lines indicate the force direction on a positive test charge, away from positive and toward negative charges.

Electric Potential

• Expressed as $V = \frac{E_p}{q}$, where V is the electric potential, $E_p$ is the electric potential energy, and q is the charge.

Electric Potential Energy

• Defined as $E_p = qV$, where $E_p$ is the electric potential energy, q is the charge, and V is the electric potential.

Electric Current

• Defined as $I = \frac{\Delta q}{\Delta t}$, where I is the current, $\Delta q$ is the charge amount, and $\Delta t$ is the time interval.

Resistance

• Defined as $R = \frac{V}{I}$, where R is the resistance, V is the voltage, and I is the current.

Ohm's Law

• Expressed as $V = IR$, where V is the voltage, I is the current, and R is the resistance.

Electrical Power

• Expressed as $P = IV$, where P is power, I is current, and V is voltage.

Descriptive Statistics: Univariate Analysis

• Univariate descriptive statistics summarizes data for a single variable.

Variable Definitions

• A variable is a characteristic that takes different values.
• A statistical unit is an item in the study population.

Types of Variables

• Qualitative variables take non-numerical values.
  • Nominal variables: categorical and unordered (e.g., eye color).
  • Ordinal variables: categorical and ordered (e.g., satisfaction level).
• Quantitative variables take numerical values.
  • Discrete variables: only finite values (e.g., number of children).
  • Continuous variables: any value within an interval (e.g., height).

Location Parameters

• Measure average position of the values in the data set

Measures of Position

• Parameters used to locate the distribution of data.
• Mean: The sum of values divided by the count: $\bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i$.
• Median: The value dividing the data distribution in half.
• Mode: The most frequent data value.

Measures of Dispersion

• Parameters used to measure the distribution of the data

Measures of Spread

• Parameters indicating data dispersion around a central point.
• Variance: The average squared difference from the mean: $S^2 = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2$.
• Standard Deviation: The square root of the variance: $S = \sqrt{S^2}$.
• Range: The difference between maximum and minimum values.
• Interquartile Range: The difference between the third and first quartiles.

Shape Parameters

• Parameters for describing the distribution's shape.
• Skewness Coefficient: Measures asymmetry.
• Kurtosis Coefficient: Measures flatness of the distribution.

Graphical Representation

• Diagrammatic representations of the data values across discrete or continuous intervals
• Bar Chart: Represents discrete variables.
• Histogram: Represents continuous variables.
• Box Plot: Shows data distribution using median, quartiles, and outliers.

Algorithmic Trading Definition

• Computer programs automatically execute trades based on instructions regarding price, timing, and quantity.

Algorithmic Trading Benefits

• Transaction costs are reduced.
• Execution speed is improved.
• Market efficiency is increased.

Order Types

• Market Order: Executes immediately at the best available price; execution is guaranteed but not the price.
• Limit Order: Executes at a specific price or better; price is guaranteed but not execution.
• Stop Order: Becomes a market order when the stop price is reached; used to limit losses or protect profits.
• Stop-Limit Order: Becomes a limit order when the stop price is reached; combines features of both stop and limit orders.

Execution Algorithms

• VWAP aims to execute at the volume-weighted average price: $VWAP = \frac{\sum_{i=1}^{n} P_i * Q_i}{\sum_{i=1}^{n} Q_i}$ Where $P_i =$ Price of trade i, $Q_i =$ Quantity of trade i
• TWAP focuses on even distribution over time: $TWAP = \frac{\sum_{i=1}^{n} P_i}{n}$ Where $P_i =$ Price at time i, $n =$ Number of prices observed
• Implementation Shortfall balances execution cost against the risk of not completing the order, minimizing difference between actual and paper portfolio prices.
• Percentage of Volume (POV) participates in trading at a pre-defined percentage of the market volume, suitable for orders needing execution without affecting market price.

High-Frequency Trading (HFT) Characteristics

• HFT involves extremely high speeds and turnover rates, using sophisticated algorithms and co-location.
• HFT Tactics include market making, arbitrage, and exploiting short-term inefficiencies.

HFT Concerns

• Potential is high for Market manipulation.
• HFT Leads to contribution to volatility of the market.
• Raises concerns over fairness and access issues.

Market Impact Definition

• The effect of a trader’s transactions on the price of an asset.

Factors Influencing Market Impact

• Order Size: Larger orders greatly impact the market.
• Market Liquidity: Less liquid markets are more sensitive.
• Order Urgency: Aggressive orders move prices.
• Information Asymmetry: Informed traders can cause larger price movements.

Market Impact Management

• Smaller Orders: Break down large orders into smaller ones.
• Passive Strategies: Use limit orders to avoid aggressive market orders.
• Timing: Execute during periods of high liquidity.
• Venue Selection: Choose venues with deep order books.

Dark Pools Definition

• Private exchanges for trading securities offer anonymity and reduced market impact.

Dark Pools Purpose

• Used by institutional Investors to trade large blocks without revealing their intentions.
• Offer Price Discovery alternative to public exchanges.

Dark Pool Concerns

• Lack of transparency.
• Raises the potential for abuse, offering certain participants unfair advantages.

Plan Definition

• A plan is a named set of configurations defining resource provisioning, instance types, AMIs, security groups, and configurations.

Reasons to Use Plans

• To codify infrastructure configurations.
• To easily reproduce configurations across multiple environments.
• To promote infrastructure as code.
• To enable version control, automation, and collaboration.

Plan Example Attributes:

• A plan block defines a named set of configurations.
• A resource block defines a specific resource to be provisioned, such as an AWS instance or security group.
• The ami attribute specifies the Amazon Machine Image (AMI) to use for the instance.
• The instance_type attribute specifies the type of instance to provision.
• The security_groups attribute specifies the security groups to associate with the instance.
• The tags attribute allows you to add metadata to the instance.
• The ingress block defines the inbound traffic rules for the security group.

How to Use Plans

• You Define your plans in HCL files.
• Then use the habitat plan apply to your infrastructure.
• Now Habitat can provision resources and they can be configured according to your specifications.