MATRIX Course

Introduction

A matrix is like an Excel table. You got columns, rows, and values. When we have a system of equations, we can easily create a matrix from it, and use methods such as GAUSS to solve the system.

We are writing $A_{n,p}$ a matrix of $n$ lines and $p$ columns. The value at the $i$-nth line and the $j$-nth column is written $a_{ij}=a_{i,j}$. The coefficients are starting from 1.

\[ \begin{split}A=\ A_{n,p}=\ A_{np}=\ \begin{pmatrix} \ a_{1,1} & \cdots & a_{1, p-1} \\ \vdots & \cdots & \vdots \\ \ a_{n-1,1} & \cdots & a_{n-1, p-1} \end{pmatrix}\end{split} \]
In R, you can create a matrix like this
# THE GOAL
#       [,1] [,2] [,3]
# [1,]    1    0    0
# [2,]    0    1    0
# [3,]    0    0    1

# special method since the matrix is special
diag(3)
# more generaly (since n=p, ommit p)
matrix(c(1,0,0,0,1,0,0,0,1), 3)
# same as (set p, useless here)
matrix(c(1,0,0,0,1,0,0,0,1), n = 3, p = 3)
# same as (byrow instead of bycolumn)
matrix(c(1,0,0,0,1,0,0,0,1), 3, 3, byrow = T)

# a (i=2, j=2)
A[2, 2]
# 1

Linear System

A linear system is a set of equations that can be solved together. Solving one will give you the value of one variable, that you can use to solve another one until you solved everything.

What's convenient is that you can use a matrix for that because the GAUSS method (and a lot of other methods in Numerical analysis) needs a matrix. The rules are that the values in the matrix are the coefficient between the variables ($0$ if not found, $1$ if $x$ or $n$ if $n * x$). The last column is the vector $b$ made with the values after the equals (second membre).


Named matrices

In French, we will call these Matrices remarquables.

The matrix you must remember is called Identity matrix (Matrice identité) $Id_n$ or $I_n$. This is a diagonal matrix having $1$ on the diagonal, which mean we have $A_n * Id_n = Id_n * A_n = A_n$. In R, you can create it with diag(n).

Zero matrix $O_{np}$ Square matrix $A_{n}$ Diagonal matrix $D_{n}$ Conjugate transpose $A^*$

Matrice nulle
All values are zero.

\[ O_{1,2}=\ \begin{pmatrix} 0&0 \end{pmatrix} \]

Matrice carré
We must have $i=j$.

\[ A_{2}=\ \begin{pmatrix} 1&2\\ 3&4\\ \end{pmatrix} \]

Matrice diagonale
Zero-Matrix with non-zero values on the diagonal.

\[ D_{2}=\ \begin{pmatrix} 1&0\\ 0&4\\ \end{pmatrix} \]

In R, call diag(c(1,4)).

Matrice conjuguée
Change the sign before the complex value $i$.
If there are no complex values, then do the transpose.

Transpose matrix $A^{T}$ Vector $A_{1,n}/A_{n,1}$ Hermitian matrix ($\mathbb{C}$)
Symmetric matrix ($\mathbb{R}$)
Unitary matrix ($\mathbb{C}$)
Orthogonal matrix ($\mathbb{R}$)

Matrice transposée
Rows become columns.

\[ A=\ \begin{pmatrix} 1&2 \end{pmatrix} \quad A^T=\ \begin{pmatrix} 1\\2 \end{pmatrix} \]

Let $A$ a matrix, in R you can transpose $A$ with t(A).

Matrice ligne/colonne or vecteur
A matrix with either one line or one column.

\[ A_{1,2}=\ \begin{pmatrix} 1&2 \end{pmatrix} \]

Matrice symétrique ($\mathbb{R}$)
Matrice hermitienne ($\mathbb{C}$)

This is a matrix equals to its transpose.

Matrice orthogonale ($\mathbb{R}$)
Matrice unitaire ($\mathbb{C}$)

This is a matrix whose inverse is equals to its transpose.

Lower triangular $L$ Strictly lower triangular $L$ Upper triangular $U$ Strictly upper triangular $U$

Matrice triangulaire inférieure (TI)
All values above the diagonal are 0.

Matrice triangulaire inférieure stricte (TIS)
All values above the diagonal are 0, including the diagonal.

Matrice triangulaire supérieure (TS)
All values below the diagonal are 0.

Matrice triangulaire supérieure stricte (TSS)
All values below the diagonal are 0, including the diagonal.


Basic operations

The basic operation you can use are


Gaussian elimination

We have a system. We are converting it to a matrix. Then, using Gauss, we get a matrix with an increasing sequence of 0 before the coefficients $p_i$. We can convert back the matrix to a system.

\[ \begin{split}\begin{cases}ax + by + ... = s_1\\ax + by + ... = s_2\\ ... \end{cases} \Leftrightarrow \begin{pmatrix}a & b & ... & | s_1 \\a & b & ... & | s_2 \\ \cdots \end{pmatrix} \Leftrightarrow^{Gauss} \begin{pmatrix}p_1 & v_1 & ... & | s_1 \\0 & p_2 & ... & | s_2 \\ \cdots \end{pmatrix} \Leftrightarrow \begin{cases}p_1 * x = s_1 + v_1 * s_2 + ... \\ p_2 * y = s_2 + ... \end{cases}\end{split} \]

Matrix inversion

The determinant of $A$, $det(A)=|A|$ is a value that is indicating if we can invert or not a matrix. A matrix is invertible (also written as $A \in Gl_n(\mathbb{R})$) if the determinant is not null, so $det(A) \neq 0$.

  • Determinant properties
    • Let $A_{1,1}=\begin{pmatrix} a \end{pmatrix}$, then we have $det(A) = a$
    • $det(AB) = det(A) * det(B)$
    • $det(A^t) = det(A)$
    • Let $A$, a diagonal matrix, $det(A)$=product of the values on the diagonal
    • The property above is working for upper/lower diagonal matrices
  • You can calculate the determinant

Then, once you have checked the determinant, you can invert a matrix using


Definiteness of a matrix

Many times, you will be asked to check the definiteness of a matrix. You can either find the eigenvalues ($\lambda$, as you can learn in the next section), or check the leading minors ($\Delta$, that you will learn in this section).

Name eigenvalues minors
positive definite
définie positive
$\forall k,\quad \lambda_k \gt 0$ $\forall k,\quad \Delta_k \gt 0$
positive semi-definite
semi-définie positive
$\forall \text{k aside $k=e$},\quad \lambda_k \gt 0,\quad \lambda_{e} = 0$ $\forall k < n,\quad \Delta_{k}>0 \text{. and } \Delta_{n}=0$
negative definite
définie négative
$\forall k,\quad \lambda_k \lt 0$ $\forall k,\quad (-1)^k \Delta_k \gt 0$
negative semi-definite
semi-définie négative
$\forall \text{k aside $k=e$},\quad \lambda_k \lt 0,\quad \lambda_{e} = 0$ $\forall k < n,\quad (-1)^k\Delta_{k}>0 \text{. and } \Delta_{n}=0$
indefinite
indéfinie
$\exists k\exists n,\quad \lambda_k \gt 0,\quad \lambda_n \lt 0$ $\Delta_{n} < 0$, and the dimension n is pairwise

Eigendecomposition of a matrix

This is called Réduction d’endomorphisme in French. The goal is to calculate $A^n$ or $exp(A)$. Our goal is to calculate some matrix so that we have $A^n = P * D^n * P^{-1}$

  • $D$: diagonal matrix, the eigenvalues of $A$ are on the diagonal
  • $P$: an invertible matrix so that we have our formula

Some vocabulary

  • Eigendecomposition: $A x = \lambda x$
  • an eigenvector (Vecteur propre): $x$
  • an eigenvalue (Valeur propre): $\lambda$
  • Spectrum of a matrix $\sigma(A)$: set of all eigenvalues
  • Rho, $\rho$ : the highest eigenvalue

You get the eigenvalues (according to Cayley–Hamilton theorem) by solving the following characteristic polynomial (Polynôme caractéristique). The polynomial's degree (degré) is $n$.

@ det(\lambda{I_n}-A_n) = 0 @


Sources