4.1. Matrices: Definitions and Operations

Matrices

Définitions Générales

• A matrix is a table of numbers, symbols, or expressions arranged in rows and columns.
• Dimension of a matrix is expressed as m x n, with m being the number of rows and n being the number of columns.
• $a_{ij}$ represents the element in the i-th row and the j-th column.
• Square matrices have equal numbers of rows and columns $m = n$
• Row matrices have only one row $m = 1$
• Column matrices have only one column $n = 1$
• Null matrices have all elements as zero.
• An identity matrix ($I_n$) is a square matrix with 1s on the main diagonal and 0s elsewhere.

Opérations Matricielles

• Addition and subtraction are possible only for matrices of the same dimension, performed element by element: $(A \pm B){ij} = a{ij} \pm b_{ij}$.
• Multiplication by a scalar involves multiplying each matrix element by the scalar: $(\lambda A){ij} = \lambda a{ij}$.
• For matrix multiplication: if A is an m x n matrix and B is an n x p matrix, the product AB is an m x p matrix with elements $(AB){ij} = \sum{k=1}^{n} a_{ik}b_{kj}$.

Transposition

• The transpose of matrix A, denoted as $A^T$, results from swapping rows and columns: $(A^T){ij} = a{ji}$.

Matrices Spéciales

• Symmetric matrices: $A^T = A$
• Anti-symmetric matrices: $A^T = -A$
• Orthogonal matrices: $A^T = A^{-1}$

Systèmes d'Équations Linéaires

Définitions

• Linear equation systems can be represented as $Ax = b$, where A is the coefficient matrix, x is the vector of unknowns, and b is the constant vector.

Méthodes de Résolution

• Gauss elimination transforms the system into an echelon form for easier solving.
• Gauss-Jordan elimination transforms the system into a reduced echelon form.
• Cramer’s rule uses determinants to solve systems with equal equations and unknowns, provided the determinant isn't zero.

Déterminants

Définition

• The determinant of a square matrix A, written as det(A) or |A|, yields a scalar calculated recursively.
• For a 2 x 2 matrix: $\det \begin{pmatrix} a & b \ c & d \end{pmatrix} = ad - bc$

Propriétés

• $\det(A^T) = \det(A)$
• $\det(AB) = \det(A) \det(B)$
• If matrix A contains a row or column of zeros, $\det(A) = 0$.
• If matrix A has two identical rows or columns, $\det(A) = 0$.

Calcul du Déterminant

• Cofactors are calculated as $C_{ij} = (-1)^{i+j}M_{ij}$, where $M_{ij}$ denotes the minor.
• Expansion by cofactors: $\det(A) = \sum_{j=1}^{n} a_{ij}C_{ij}$ (expansion by the i-th row).

Espaces Vectoriels

Définitions

• A vector space is a set with addition and scalar multiplication that satisfies associativity, commutativity, existence of neutral and inverse elements, among other properties.
• Vector Subspaces are subsets of a vector space, which are themselves vector spaces.
• Linear Combination involves multiplying vectors by scalars and adding them.
• Linear Independence means that the only way to get a null vector is to set all scalars to zero.
• Bases are sets of linearly independent vectors generating the vector space.
• Dimension is defined by the number of vectors in a base.

Applications Linéaires

• An application $f: E \rightarrow F$ (with E and F being vector spaces) is linear if $f(u + v) = f(u) + f(v)$ and $f(\lambda u) = \lambda f(u)$.
• Kernel definition: $\text{ker}(f) = {u \in E \mid f(u) = 0}$
• Image definition: $\text{im}(f) = {f(u) \mid u \in E}$

Valeurs Propres et Vecteurs Propres

Définitions

• Eigenvalues are scalars $\lambda$ satisfying $Av = \lambda v$ with a non-zero vector v.
• Eigenvectors are non-zero vectors v that satisfy $Av = \lambda v$ for an eigenvalue $\lambda$.
• The characteristic polynomical is expressed as $\det(A - \lambda I)$

Diagonalisation

• Matrix A is diagonalizable given an invertible matrix P and a diagonal matrix D, such that $A = PDP^{-1}$. The columns of P represent the eigenvectors of A, and the diagonal elements of D are the corresponding eigenvalues.

Radiative Processes

What is Radiation

• Any object with temperature $T > 0$ emits radiation
• Radiation is energy transfer via photons.
• It is an essential energy transfer mechanism in astrophysics.

Blackbody Radiation

• A blackbody absorbs all radiation.
• It emits radiation at all wavelengths.
• Its emission relies only on its temperature $T$
• A perfect emitter

Specific Intensity

• $I_{\nu}$: energy flux per unit frequency per unit solid angle.
• $I_{\nu} = \frac{dE}{dt d\nu dA d\Omega}$
• Units: [erg s$^{-1}$ Hz$^{-1}$ cm$^{-2}$ ster$^{-1}$]
• $dE$ is the amount of energy
• $dt$ is the time interval
• $d\nu$ is the frequency interval
• $dA$ is the area
• $d\Omega$ is the solid angle

Planck Function

• Specific intensity of blackbody:
• $B_{\nu}(T) = \frac{2h\nu^{3}}{c^{2}} \frac{1}{e^{\frac{h\nu}{kT}}-1}$
  • $h$ is Planck's constant ($6.626 \times 10^{-27}$ erg s)
  • $c$ is the speed of light ($3 \times 10^{10}$ cm/s)
  • $k$ is Boltzmann's constant ($1.381 \times 10^{-16}$ erg/K)
  • $T$ is the temperature in Kelvin

Wien's Displacement Law

• Wavelength at Planck function's maximum:
• $\lambda_{max} = \frac{b}{T}$
• $b \approx 0.29$ cm K (Wien's displacement constant)

Stefan-Boltzmann Law

• Total energy emitted per unit area:
• $F = \sigma T^{4}$
• $\sigma = \frac{2\pi^{5}k^{4}}{15c^{2}h^{3}} = 5.67 \times 10^{-5}$ erg s$^{-1}$ cm$^{-2}$ K$^{-4}$ (Stefan-Boltzmann constant)

Luminosity

• Luminosity of spherical blackbody with radius R and temperature T:
• $L = 4\pi R^{2} \sigma T^{4}$

Caveats

• Real objects are not blackbodies
• Emission relies on frequency
• Need emissivity, $\epsilon_{\nu}$: $0 \le \epsilon_{\nu} \le 1$
• $I_{\nu} = \epsilon_{\nu} B_{\nu}(T)$

Learn about matrices, their dimensions, and specific types such as square, row, and column matrices. Understand matrix operations like addition, subtraction, and scalar multiplication. Explore the requirements for performing these operations.