Special Types of Matrices

Matrices

• Matrices are mathematical objects organized in rows and columns.

Special Matrices

• Certain matrices have unique properties that make them useful in various applications.

Zero Matrix

• Every element in a zero matrix is zero.

Identity Matrix

• An identity matrix is a square matrix with ones on the diagonal and zeros everywhere else.
• Multiplication of a matrix A by the identity matrix results in A.

Diagonal Matrix

• Only the diagonal elements of a diagonal matrix have non-zero values.

Transpose Matrix

• Rows and columns are swapped to form the transpose.
• Denoted as $A^T$.

Symmetric Matrix

• A matrix equal to its transpose is symmetric.

Skew-Symmetric Matrix

• The transpose is equal to the negative of the matrix ($A^T = -A$).
• Diagonal elements are zero.

Triangular Matrix

• There are upper and lower variants.

Upper Triangular Matrix

• All elements below the main diagonal are zero.

Lower Triangular Matrix

• All elements above the main diagonal are zero.

Matrix Operations

• Matrices can undergo several operations, provided size constraints are met.

Addition

• Corresponding elements are added between equally sized matrices.
• $A + B = [a_{ij} + b_{ij}]$

Subtraction

• Corresponding elements are subtracted.
• $A - B = [a_{ij} - b_{ij}]$

Scalar Multiplication

• Each element is multiplied by the scalar.
• $cA = [ca_{ij}]$

Matrix Multiplication

• For A ($m \times n$) and B ($n \times p$), the product AB is $m \times p$.
• Element (i,j) of AB is the sum of the products of the i-th row of A and the j-th column of B.
• $(AB){ij} = \sum{k=1}^{n} a_{ik}b_{kj}$

Algorithmes de Hachage (Hashing Algorithms)

Hashing yields a fixed-size output (hash) from a variable-size input.

Propriétés (Properties)

• Deterministic: Identical inputs produce identical hashes.
• Rapid computation.
• Preimage resistance: Infeasible to find an input matching a given hash.
• Second preimage resistance: Computationally difficult to discover a distinct second input that generates the same hash as a specific input.
• Collision resistance: Computationally infeasible to locate two different inputs yielding the same hash.
• One-way function: Calculation is easy in one direction, hard to reverse.
• Avalanche effect: Minor input change radically alters the hash.

Applications

• Used for verifying data integrity, password storage, hash tables, digital signatures, and blockchain technology.

Exemples (Examples)

• Includes MD5, SHA-1, SHA-256, SHA-384, SHA-512, SHA-3, bcrypt, scrypt, and Argon2.
• MD5 and SHA-1 are considered obsolete because of security vulnerabilities.

Attaques Courantes (Common Attacks)

• Force brute (brute-force), dictionnaire (dictionary), arc-en-ciel (rainbow table), collision, and longueur d'extension (length extension) attacks.
• Countermeasures include salting passwords and selecting strong hash functions.

Sélection (Selection)

• Prioritize security and stardardization.
• Standard algorithms ensure interoperability.
• Evaluation requires taking into account overall performance, collision resistence, hash size and availability.

Conclusion

• Hashing algorithm implementation necessitates considering properties, applications, and limitations for efficacious utilization.

Bernoulli's Principle

• As the speed of a fluid increases, the pressure decreases (or the potential energy decreases).
• Only applies to isentropic flows where irreversible processes are negligible.

Incompressible Flow Equation

• $P + \frac{1}{2} \rho v^2 + \rho g h =$ constant
• P is the static pressure, $\rho$ is the density, v is the flow speed, g is the acceleration due to gravity, and h is the elevation.

Venturi Meter

• The Venturi effect refers to the reduction of fluid pressure when a fluid flows through a constricted section of a pipe.

How it Works

• Fluid velocity increases and static pressure decreases as fluid moves through a converging section.
• Consists of converging, throat and diverging sections.
• Pressure is minimal in the throat.
• Flow rate can be determined by measuring the pressure difference between the inlet and the throat.

Applications

• The Venturi meter has usage in measuring flow rates in pipelines, carburetors and aerodymanics.

Pitot Tube

• Used to measure fluid flow velocity.

How it Works

• A tube points directly into the fluid flow and measures the stagnation pressure.
• Stagnation pressure is used to determine flow velocity.

Calculating Flow Velocity

• $v = \sqrt{\frac{2(P_t - P_s)}{\rho}}$
• To measure static pressure $P_s$, a static port is needed.

Applications

• Measuring airspeed of aircraft, air/gas velocities in industrial applications, and water flow.

Resúmenes (Summaries)

¿Qué es un resumen? (What is a summary?)

• A summary is a concise presentation of the most important points of a text.