Matrices and graphs: senior math course in PDF.

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1 I. The notion of matrix.
2 II. Graphs.
3 III. Application of matrix calculus to graphs.

the chapter on matrices is very instructive and requires some concentration on the part of the student. This course will allow him to develop new skills.

I. The notion of matrix.

Let m, n and p be three non-zero natural numbers.

1. notion of matrix and operations.

Definition:

A matrix of size (or format) $n\times \,p$ is an array of real numbers n rows and p columns.

Example:

matrix

is a matrix with 2 rows and 3 columns and therefore of size $2\times \,3$ .

Definitions:

When n = 1, we say that M is a row matrix, formed by a single row.
Then, when p = 1, we say that M is a column matrix, formed by a single column.
And when n = p, we say that M is a square matrix of order n.
A diagonal matrix is a square matrix, whose terms are all zero except
when i=j.
The identity matrix of order n is the diagonal matrix of order n whose coefficients
diagonals are equal to 1. It is noted I_n .
The null matrix of size $n\times \,p$ , denoted $O_{n,p}$ , is the matrix of size $n\times \,p$ , whose coefficients are all zero.

Examples:

Definition:

Two matrices A and B of size $n\times \,p$ are equal when, for any $i\in\,\,\{\,1,2,...,n\,\,\}$ and $j\in\,\,\{\,1,2,...,p\,\,\}$

we have $(a_{ij})=(b_{ij})$ .

Definition:

A square matrix of order is symmetric when, for any $i\in\,\,\{\,1,2,...,n\,\,\}$ and $j\in\,\,\{\,1,2,...,n\,\,\}$

we have $(a_{ij})=(a_{ji})$ .

Example:

The following matrix is symmetric.

matrix

2. Operations on matrices.

Definitions:

Let $A\,=(a_{ij})$ and $B\,=(b_{ij})$ be two matrices of size $n\times \,p$ .

Thesum of the matrices A and B, noted A + B, is the matrix $C\,=(c_{ij})$ of size $n\times \,p$
such that, for all $1\leq\,\,i\leq\,\,n$ and $1\leq\,\,j\leq\,\,p$ , we $c_{i,j}=a_{i,j}+b_{i,j}$ .
The product of the matrix A by a real $\lambda$ , noted $\lambda\,A$ , is the matrix $M\,=\,(m_{ij})$ of
size $n\times \,p$ such that, for all $1\leq\,\,i\leq\,\,n$ and $1\leq\,\,j\leq\,\,p$ , we have $m_{i,j}=\lambda\times \,a_{i,j}$ .

Properties:

Let A, B and C be three matrices of the same size and $\alpha$ and $\beta$ two real numbers.

A+ B = B + A (commutativity of the sum of matrices)
A+(B +C) = (A+ B)+C (associativity of the sum of matrices)
$1\times \,A=A\times \,1=A$
$(\alpha\,+\beta\,)A=\alpha\,A+\,\beta\,A$
$\alpha\,(A+B)=\alpha\,A+\alpha\,B$

Definition:

We call the matrix M=(-1)A , noted , the opposite matrix of A such that,

for all $1\leq\,\,i\leq\,\,n$ and $1\leq\,\,j\leq\,\,p$ , we have $m_{i,j}=-a_{i,j}$ .

Furthermore, we note A-B the matrix A+(-B) .

Examples:

Definition:

Let $L=(l_{1,1},l_{1,2},...,l_{1,n})$ be a row matrix of size $1\times \,n$ and

column matrix

a column matrix of size column matrix of size $n\times \,1$ .

Then the product $L\times \,C$ is the real number defined by :

Example:

3.product of two matrices.

Definition:

If A is a matrix of size $m\,\times \,n$ and B a matrix of size $n\times \,p$ , the product of
matrices A and B, noted $A\times \,B$ or , is the matrix $C\,=\,(c_{i,j})$ of size $m\times \,p$ such that,
for all $1\leq\,\,i\leq\,\,n$ and $1\leq\,\,j\leq\,\,p$ , we have $c_{i,j}=\sum_{k=1}^{n}b_{i,k}\times \,b_{k,j}$ .
In other words, the element $c_{i,j}$ . is the product of the i-th line of A by the j-th
B column.

Properties:

Let A, B and C be three matrices and $\lambda$ a real number.

Subject to the definition of products and sums, we have :

$(A\times \,B)\times \,C=A\times (\,B\times \,C)=A\times \,B\times \,C$
$A\times \,(B+C)=A\times \,B+\,A\times \,C$
$(A+B)\times \,C=A\times \,C+\,B\times \,C$
$(\lambda\,A)\times \,B=A\times \,(\lambda\,B)=\lambda\,AB$
$I_n\times \,A=A\times \,I_n=A$

Definition:

Let A be a square matrix of order n and k a nonzero natural number.
The k-th power of A, denoted A^k , is the matrix $A^k=A\times \,A\times \,...\times \,A$ (k times ).

4.invert of a matrix and system resolution.

Definition:

A square matrix A of size n is invertible when there is a square matrix B
of size n such that $A\times \,B\,=\,B\,\times \,A=I_n$ .

Definition:

Let be a square matrix of order 2.

The determinant of A is the real, denoted det(A), defined by .

Ownership:

A square matrix is invertible if, and only if, its determinant is non-zero.
In particular, if is invertible then

Ownership:

Let A be a square matrix of size n and X and B two column matrices with n rows.
If A is invertible, then the matrix writing system AX = B admits a unique
solution given by the column matrix $X\,=\,A^{-1}\times \,B$ .

Example:

II. Graphs.

Definitions:

A graph is a representation composed of vertices (points) connected by
edges (segments).
A directed graph is a graph whose edges have a direction of travel.
Theorder of a graph is the number of vertices of this graph.
The degree of a vertex is the number of edges incident to this vertex, without taking into account
their possible direction of travel.

Example:
The graph below is of order 5.
The vertices K and L are of degree 3.
The vertices M , M and M, are of degree 2.

graph

Definitions:

Two vertices are adjacent when they are connected by at least one edge.
A graph is complete when all its vertices are adjacent to each other.

Examples:
1. The graph below is complete because all its vertices are two by two adjacent.

graph
2. The graph below is not complete because the vertices A and B, for example, are not
not adjacent.

graph

Definition:

For an undirected graph, a chain is a sequence of consecutive edges connecting
two vertices (possibly merged).
The length of a chain is the number of edges in it.
For a directed graph, a path is a sequence of consecutive edges connecting two
vertices (possibly merged) taking into account the direction of travel of the edges.

graph

III. Application of matrix calculus to graphs.

Definition:

Let n be a non-zero natural number. Consider a graph of order n (directed or not)
whose vertices are numbered from 1 to n, then arranged in ascending order.
The adjacency matrix of this graph is the square matrix of size n, whose

coefficient $a_{i,j}$ is equal to the number of edges starting from vertex i to arrive at vertex j.

Example:

By noting M the adjacency matrix of the graph below obtained by arranging

the vertices in alphabetical order.

We have:

matrix matrix

Ownership:

Let n and k be two non-zero natural numbers and M the adjacency matrix of a graph
of order n, whose vertices are numbered from 1 to n and arranged in ascending order
The term of the i-th row and the j-th column of the matrix M^k gives the
number of chains (or paths) of length k connecting vertex i to vertex j.

Example:

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Matrices and graphs: senior math course in PDF.

I. The notion of matrix.

1. notion of matrix and operations.

2. Operations on matrices.

3.product of two matrices.

4.invert of a matrix and system resolution.

II. Graphs.

III. Application of matrix calculus to graphs.

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