Diagonalizacion

+D=FJAH 
Matrix calculus and applications.

The goals of this chapter are the followings: 1. To review some general concepts about vector spaces and matrix calculus. 2. To calculate the characteristic polynomial, the characteristic equation, the eigenvalues and the eigenvectors of a square matrix. 3. To calculate the diagonalizing matrix and the diagonal matrix related to a square matrix. 4.To study the orthogonal diagonalization of a symmetric matrix. 5. To implement all the concepts in the CAS (Computer sary to upload the package linalg:
Input > with(linalg);
Algebra System)

Maple V. It will be neces-

1.1

Introduction

1.1.1 Vector spaces
Denition 1 A eld (F, +, ×) is a set endowed with two inner laws, an addition + and a multiplication ×,
verifying the followingaxioms: i) Addition axioms: a) Commutativity: a + b = b + a, ∀a, b ∈ F . b) Associativity: (a + b) + c = a + (b + c), ∀a, b, c ∈ F . c) Identity element: There exists an element 0 ∈ F such that 0 + a = a + 0 = a, ∀a ∈ F. d) Inverse element: Given a ∈ F , there exists −a ∈ F such that a + (−a) = (−a) + a = 0. ii) Multiplication axioms: a) Commutativity: a × b = b × a, ∀a, b ∈ F . b) Associativity:(a × b) × c = a × (b × c), ∀a, b, c ∈ F . c) Identity element: There exists an element 1 ∈ F such that 1 × a = a × 1 = a, ∀a ∈ F. d) Inverse element: Given a ∈ F , there exists a−1 ∈ F such that a × a−1 = a−1 × a = 1.
Applied Mathematics for Building Construction I's Notes. Technical Architecture School. University of Seville.
1 C-I group. Academic Year 2010 - 2011. Lecturer: Raúl Manuel FalcónGanfornina.

Page 2 iii) Addition and multiplication axioms:

CHAPTER 1. MATRIX CALCULUS AND APPLICATIONS.

a) Distributivity: (a + b) × c = a × b + a × c, ∀a, b, c ∈ F .

An example of eld is given by the set of real numbers R, with the usual sum + and product ×.

Denition 2 Given a eld (F, +, ×), a F -vector space (V, +, ·) is a set V whose elements are called vectors
(written inbold type) and which is endowed with two laws, a vector addition + and a scalar multiplication · between vectors and scalars of the eld F , verifying the following properties: i) Vector addition: a) b) c) d) Commutativity: v1 + v2 = v2 + v1 , ∀v1 , v2 ∈ V . Associativity: (v1 + v2 ) + v3 = v1 + (v2 + v3 ), ∀v1 , v2 , v3 ∈ V . Identity element: There exists an element 0 ∈ V , such that 0 + v = v +0 = v, ∀v ∈ V . Inverse element: Given v ∈ V , there exists −v ∈ V such that v + (−v) = (−v) + v = 0.

ii) Scalar multiplication: a) Identity element: 1 · v = v, ∀v ∈ V . iii) Vector addition and scalar multiplication: a) Associativity: a · (b · v) = (a × b) · v, ∀a, b ∈ F and ∀v ∈ V . b) Distributivity of scalar multiplication: (a + b) · v = a · v + b · v, ∀a, b ∈ F and ∀v ∈ V . c)Distributivity of vector addition: a · (v1 + v2 ) = a · v1 + a · v2 , ∀a ∈ F and ∀v1 , v2 ∈ V .

Maple Command vector:
Input > v:=vector([v1,v2,v3]); Output > v := [v1, v2, v3]

Denition 3 Let (F, +, ×) be a eld and let (V, +, ·) be a F -vector space. It is said that m vectors of V ,
v1 , v2 , ..., vm are linearly independent if, for all λ1 , λ2 , ..., λm ∈ F :
m ∑ i=1

λi · vi = 0 ⇒ λ1 = λ2 = ...= λm = 0.

Given a set S = {v1 , v2 , ..., vm } of m vectors of V , the rank of S (rk(S)) is the number of linearly independent vectors of S . The vector space generated by S is the dened as:
{ Span(S) = v ∈ V : ∃λ1 , λ2 , ..., λm ∈ F such that v =
m ∑ i=1

} λi · v i .

A basis of V is any set of linearly independent vectors which generates V . Any basis of V has the same number ofvectors, which is called the dimension of V . Finally, if {v1 , v2 , ..., vn } is a basis of V , then, given ∑ a vector v ∈ V , if v = n λi · vi , then the vector v can be represented by the coordinates (λ1 , λ2 , ..., λn ). i=1

Applied Mathematics for Building Construction I's Notes. Technical Architecture School. University of Seville.

1 C-I group. Academic Year 2010 - 2011. Lecturer: Raúl...