Introduction to Matrices

Matrices

• Matrices are fundamental mathematical objects that organize numbers, symbols or expressions in a rectangular array and are used in various fields for problem solving.

Definition of a Matrix

• A matrix is a rectangular array of elements arranged in rows and columns, denoted by a capital letter and enclosed in brackets or parentheses.
• $A$ is a matrix with $m$ rows and $n$ columns, where $a_{ij}$ is the element in the $i$-th row and $j$-th column.
• The order of a matrix is $m \times n$ if it has $m$ rows and $n$ columns.
• Each entry in a matrix is called an element.

Types of Matrices

• Row matrix: A matrix with only one row and any number of columns.
  • Example: $A = \begin{bmatrix} 1 & 2 & 3 \end{bmatrix}$ is a row matrix of order $1 \times 3$.
• Column matrix: A matrix with only one column and any number of rows.
  • Example: $B = \begin{bmatrix} 4 \ 5 \ 6 \end{bmatrix}$ is a column matrix of order $3 \times 1$.
• Square matrix: A matrix with an equal number of rows and columns.
  • Example: $C = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}$ is a square matrix of order $2 \times 2$.
• Diagonal matrix: A square matrix where all elements outside the main diagonal are zero.
  • Example: $D = \begin{bmatrix} 5 & 0 & 0 \ 0 & -2 & 0 \ 0 & 0 & 3 \end{bmatrix}$ is a diagonal matrix of order $3 \times 3$.
• Scalar matrix: A diagonal matrix in which all the diagonal elements are equal.
  • Example: $E = \begin{bmatrix} 7 & 0 \ 0 & 7 \end{bmatrix}$ is a scalar matrix of order $2 \times 2$.
• Identity matrix: A scalar matrix with all diagonal elements equal to 1, denoted as $I_n$ where $n$ is the order.
  • Example: $I_2 = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}$ is an identity matrix of order 2.
• Null matrix: A matrix with all elements equal to zero, denoted as $O_{m \times n}$.
  • Example: $O_{2 \times 3} = \begin{bmatrix} 0 & 0 & 0 \ 0 & 0 & 0 \end{bmatrix}$ is a null matrix of order $2 \times 3$.

Operations on Matrices

• Addition of Matrices: Two matrices of the same order $m \times n$ can be added together; the sum $C = A + B$ is computed by adding corresponding elements: $c_{ij} = a_{ij} + b_{ij}$.
  • Example: $A + B = \begin{bmatrix} 1+5 & 2+6 \ 3+7 & 4+8 \end{bmatrix} = \begin{bmatrix} 6 & 8 \ 10 & 12 \end{bmatrix}$
• Subtraction of Matrices: Two matrices of the same order $m \times n$ can be subtracted; the difference $C = A - B$ is computed by subtracting corresponding elements: $c_{ij} = a_{ij} - b_{ij}$.
  • Example: $A - B = \begin{bmatrix} 9-1 & 10-2 \ 11-3 & 12-4 \end{bmatrix} = \begin{bmatrix} 8 & 8 \ 8 & 8 \end{bmatrix}$
• Scalar Multiplication: Multiplying a matrix $A$ of order $m \times n$ by a scalar $k$ results in a new matrix $kA$ where each element is the product of $k$ and the corresponding element of $A$: $kA = [ka_{ij}]$.
  • Example: $2A = \begin{bmatrix} 2 \times 1 & 2 \times 2 \ 2 \times 3 & 2 \times 4 \end{bmatrix} = \begin{bmatrix} 2 & 4 \ 6 & 8 \end{bmatrix}$

Properties of Matrix Operations

• Commutative Law: $A + B = B + A$
• Associative Law: $(A + B) + C = A + (B + C)$
• Existence of Additive Identity: $A + O = A = O + A$, where $O$ is a null matrix.
• Existence of Additive Inverse: $A + (-A) = O = (-A) + A$, where $-A$ is the additive inverse of $A$.

Properties of Scalar Multiplication

• Distributive Law over Matrix Addition: $k(A + B) = kA + kB$
• Distributive Law over Scalar Addition: $(k + l)A = kA + lA$
• Associative Law: $k(lA) = (kl)A = l(kA)$
• Multiplicative Identity: $1A = A$

Quantum Mechanics

• Quantum mechanics is a branch of physics that deals with the behavior of matter and energy at the atomic and subatomic levels.

History

• 1900: Max Planck introduces the concept of quantization of energy to explain blackbody radiation.
• 1905: Albert Einstein explains the photoelectric effect by photons.
• 1913: Niels Bohr develops a model of the atom with quantized energy levels.
• 1924: Louis de Broglie proposes that matter has wave-like properties.
• 1925: Werner Heisenberg formulates matrix mechanics.
• 1926: Erwin Schrödinger develops wave mechanics.
• 1927: Werner Heisenberg formulates the uncertainty principle
• 1928: Paul Dirac develops a relativistic theory of quantum mechanics.

Core concepts include

• Quantization - energy, momentum, etc, take on discrete values.
• Wave-particle duality
• Superposition
• Uncertainty Principle - there is a fundamental limit to the accuracy
• Entanglement - entanglement links systems and share the same fate

Key equations within quantum mechanics

• Planck's equation: $E = h\nu$ where E is energy, h is Planck's constant, and $\nu$ is frequency
• de Broglie's equation: $\lambda = \frac{h}{p}$ where $\lambda$ is wavelength, h is Planck's constant, and p is momentum.
• Schrödinger equation: $i\hbar\frac{\partial}{\partial t}\Psi = \hat{H}\Psi$ where $i$ is the imaginary unit, $\hbar$ is the reduced Planck constant, $\Psi$ is the wave function, t is time and $\hat{H}$ is the Hamiltonian operator.

Quantum mechanics has numerous applications

• Laser
• Transistors
• Medical imaging
• Quantum computing

Branches of Quantum Mechanics

• Quantum Field Theory (QFT): Combines quantum mechanics with special relativity
• Quantum Optics: Interactions between light and matter at quantum level
• Quantum Information Theory: Studies storage, processing, and transmission of information using quantum systems
• Quantum Gravity: Theoretical framework that attempts to unify quantum mechanics with general relativity

Challenges facing quantum mechanics

• Interpretation: There is no single, universally accepted interpretation of quantum mechanics.
• Quantum Gravity:
• Scalability: Building large-scale quantum computers is a difficult technological challenge.

Transistor Parameters

• $\beta$ Common-emitter current gain
• $I_S$ Saturation current
• $V_{BE(on)}$ - Base-emitter on voltage
• $V_A$ Early voltage
• $k_n, k_p$: Transconductance parameter in MOSFETs
• $V_t$ Threshold voltage
• $\lambda$: Channel-length modulation parameter

Algorithmic Trading

• Algorithmic trading is combining mathematics, statistics and computer science
• Understanding theoretical foundations of algorithmic trading
• Design implement and backtest trading strategies
• Evaluating performance of trading algorithms
• Manage risk
• Deploy algoritms in a live trading environment

Course Schedule

• Introduction to Algorithmic Trading
• Market Microstructure and Order Book Dynamics
• Backtesting Methodologies and Performance Evaluation
• Risk Management for Algorithmic Trading
• Machine Learning for Trading
• High-Frequency Trading and Latency Optimization
• Portfolio Construction and Optimization
• Execution Strategies and Order Placement
• Regulation and Compliance in Algorithmic Trading
• Advanced Topics in Algorithmic Trading

Assessment

• Homework Assignments: 30%
• Midterm Exam: 30%
• Final Project: 40%

Préparation à l'agrégation de mathématiques

Suites et séries de fonctions

Exercice 1

• Study the convergence of the following series of functions:
  • $\sum_{n \geq 1} \frac{x^{n}}{n}, x \in[0,1[$
  • $\sum_{n \geq 1} \frac{x^{n}}{n^{2}}, x \in[-1,1]$
  • $\sum_{n \geq 1} \frac{\sin (n x)}{n^{2}}, x \in \mathbb{R}$
  • $\sum_{n \geq 1} \frac{x}{n\left(1+n x^{2}\right)}, x \in \mathbb{R}$
  • $\sum_{n \geq 1} \frac{e^{-n x}}{n}, x \in \mathbb{R}_{+}$
  • $\sum_{n \geq 1} x^{n} \sin (n x), x \in]-1,1[$
  • $\sum_{n \geq 1} \frac{\cos (n x)}{n^{\alpha}}, x \in \mathbb{R}$ ( $\alpha>1$ fixé)
  • $\sum_{n \geq 1} \frac{(-1)^{n}}{x+n}, x \in \mathbb{R}_{+}$

Exercice 2

• Consider the function $\zeta(x)=\sum_{n=1}^{+\infty} \frac{1}{n^{x}}$ for $x>1$
  • Show that $\zeta$ is of class $C^{\infty}$ on $] 1,+\infty[$.
  • Calculate $\zeta^{\prime}(x)$ for $x>1$.
  • Show that $\zeta$ is convex.

Function Convexe

• A function $f: I \rightarrow \mathbb{R}$ is convex on $I$ if for every $x, y \in I$ and every $t \in[0,1]$, we have $f(t x+(1-t) y) \leq t f(x)+(1-t) f(y)$

Singular Value Decomposition (SVD)

• For any matrix $A \in \mathbb{R}^{m \times n}$, there exist orthogonal matrices $U \in \mathbb{R}^{m \times m}$, $U^T U = I$ and $V \in \mathbb{R}^{n \times n}$, $V^T V = I$

Steps in computing SVD

• calculate the SVD
• Find eigenvalues
• $\sigma_i^2$ are eigenvalues of $AA^T$
• Solve for $\lambda^2$
• Find eigenvectors
• Find U

Fourier Transform Properties

• $\mathcal{F} {af(t) + bg(t)} = aF(\omega) + bG(\omega)$
• $\mathcal{F} {f(at)} = \frac{1}{|a|} F(\frac{\omega}{a})$ Compression in time expansion in frequency, Expansion in time compression in frequency

Time Shifting

• $\mathcal{F} {f(t - t_0)} = e^{-j\omega t_0} F(\omega)$

Frequency Shifting

• $\mathcal{F} {e^{j\omega_0 t} f(t)} = F(\omega - \omega_0)$

Convolution

• $\mathcal{F} {f(t) * g(t)} = F(\omega) G(\omega)$ is defined

• $\alpha$ and $\beta$ are tautology.

• $\alpha$ or $\beta$ are lógically equivalent

Leyes de Equivalencia

• Laws to transform logic

Reglas de Inferencia

• Rules to define logic