Table of Contents
Table of Contents
“Exercises of Differential Linear Systems”
INTRODUCTION
OUTLINE OF MATRIX THEORY
2x2 LINEAR DIFFERENTIAL SYSTEMS
3x3 LINEAR DIFFERENTIAL SYSTEMS
“Exercises of Differential Linear Systems”
“Exercises of Differential Linear Systems”
SIMONE MALACRIDA
In this book exercises are carried out regarding the following mathematical topics:
2x2 and 3x3 linear differential systems
Cauchy problems related to linear systems with constant coefficients
search for and determination of eigenvalues related to linear systems
Initial theoretical hints concerning matrix theory are also presented to make the performance of the exercises understandable.
Simone Malacrida (1977)
Engineer and writer, has worked on research, finance, energy policy and industrial plants.
ANALYTICAL INDEX
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INTRODUCTION
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I – OUTLINE OF MATRIX THEORY
Definitions
Operations and properties
Matrix calculation
Applications
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II – 2x2 LINEAR DIFFERENTIAL SYSTEMS
Exercise 1
Exercise 2
Exercise 3
Exercise4
Exercise 5
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III - 3x3 LINEAR DIFFERENTIAL SYSTEMS
Exercise 1
Exercise 2
Exercise 3
Exercise 4
Exercise 5
Exercise 6
Exercise 7
Exercise 8
INTRODUCTION
INTRODUCTION
In this workbook some examples of calculations related to linear differential systems and related Cauchy problems are carried out.
Furthermore, an introduction to matrix theory is presented, necessary to understand the method of solving linear differential systems.
These systems play a primary role within the broader casuistry of differential systems. In fact, numerical analysis tends to bring every type of calculation back to a linear system, by adopting appropriate linearization procedures.
What is presented in this workbook is generally covered in advanced mathematical analysis courses (analysis 2) and therefore specific knowledge of integral calculus and of what is inherent in basic mathematical analysis courses is required.
I
OUTLINE OF MATRIX THEORY
OUTLINE OF MATRIX THEORY
To understand the resolution of linear systems, it is necessary to introduce some concepts related to matrix notation.
This will be useful in carrying out the exercises proposed in the next two chapters.
Definitions
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A matrix is a table of items sorted by rows and columns .
Given m rows and columns, the matrix is called "m times n" and is denoted by a capital letter.
Each element of the matrix is denoted by two subscripts, the first indicating the row, the second the column.
Vectors can be considered matrices in simplified form, having only one row or one column.
A matrix of dimension 1xn is called a row matrix , if instead it is mx1 it is called a column matrix .
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Operations and properties
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The sum and the difference between matrices having the same dimension is given by the sum of the single elements.
Multiplication by a scalar is done by multiplying each individual element by the scalar.
Multiplication between matrices is carried out in "rows by columns" form and can only be done if the number of columns of the first matrix is equal to the number of rows of the second matrix and we have these formulas for the product :
which is a generalization of the dot product between vectors.
This operation is not commutative, while all other properties of the product and the sum are preserved.
We define 0 as the null matrix made up of zeros only, while the opposite of a matrix is given by the matrix having all the elements multiplied by -1 .
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The following properties hold for the sum:
Property of existence of the neutral element A+0=0+A=A
Property of existence of the opposite element A+(-A)=0
Associative property (A+B)+C=A+(B+C)
Commutative property A+B=B+A
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The following properties hold for the product with a scalar:
Property of existence of the neutral element 1A=A
Associative property (ab)A=a(bA)
Distributive property a(A+B)=aA+aB
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The following properties hold for the product of matrices:
Associative property (AB)C=A(BC)
Distributive property (A+B)C=AC+BC
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The product of matrices also generalizes the product of a matrix and a vector.
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Matrix calculation
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The matrix obtained by exchanging the rows and columns is called transposed.
We have the following relations and properties:
The matrix obtained by taking the complex conjugate of its elements is called conjugate.
A conjugate transposed matrix is defined as the matrix obtained by transposing and then conjugating the elements.
It can be seen that the two operations commute with each other:
Moreover:
Impressum
Verlag: BookRix GmbH & Co. KG
Tag der Veröffentlichung: 20.04.2023
ISBN: 978-3-7554-3965-3
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