Vectors and Matrices Basilio Bona DAUIN Politecnico di Torino - PowerPoint PPT Presentation

Vectors and Matrices Basilio Bona DAUIN – Politecnico di Torino October 2013 B. Bona (DAUIN) Vectors and Matrices October 2013 1 / 40

1. Geometrical vectors A geometrical vector p represents a point P in space. The point P is an abstraction that often, but not always, requires a representation. Vector representations are given wrt a reference frame (see below) If P ∈ R 3 (a 3D space) then     p x p 1 p a ∈ R 3 = p y ≡ p 2     p z p 3 a a p i is the vector i -th coordinate wrt the chosen reference frame R a . B. Bona (DAUIN) Vectors and Matrices October 2013 2 / 40

Affine geometry To treat points as vectors in linear (vector) spaces implies the definition of a zero point (the origin), i.e., a point with particular privileged characteristics. Since, in many applications, this is not always the case, a particular geometry that is “origin-free” has been set up. It is called affine geometry and is defined on affine spaces. Affine geometry is at the base of projective geometry and perspective transforms, as well as homogeneous vectors. It will not be considered in the present context; the interested reader can find more details in [05]. B. Bona (DAUIN) Vectors and Matrices October 2013 3 / 40

2. Physical vectors A physical vector − → QP represents a physical quantity, such as linear or angular velocity, gravitational acceleration, force, torque, etc. A physical vectors is an oriented segment (aka directed line segment), with an application point Q that can be free or constrained, a direction and a magnitude. − → QP may be reference-free (as, e.g., a physical entity independent from the way we represent it, and from an origin), but we are usually representing it by a vector     p 1 − q 1 v 1 v qp ∈ R 3 = p − q =  ≡ p 2 − q 2 v 2    p 3 − q 3 v 3 that is the difference ot two geometrical vectors p ↔ P q ↔ Q Example: the local gravitational acceleration has direction and magnitude that are “absolute”, since they do not depend on the reference frame chosen, but on the space-time geometry. B. Bona (DAUIN) Vectors and Matrices October 2013 4 / 40

Physical vectors Figure: Physical vectors. B. Bona (DAUIN) Vectors and Matrices October 2013 5 / 40

Physical vectors We represents physical vectors with an icon; the most used icon is an arrow. Figure: The vector icon. This icon is sometimes misleading, as we will see, considering the symmetry properties of the physical vectors. There are two types of physical vectors: polar and axial ones. B. Bona (DAUIN) Vectors and Matrices October 2013 6 / 40

Polar vectors Polar vectors (see [01]) are physical vectors that are symmetrical wrt a reflection through a parallel plane, and are skew-symmetrical wrt a reflection through a perpendicular plane. Examples of physical polar vectors are displacements, linear velocities and forces. Figure: Polar vector. B. Bona (DAUIN) Vectors and Matrices October 2013 7 / 40

Axial vectors Axial vectors are physical vectors that are skew-symmetrical wrt a reflection through a parallel plane, and are symmetrical wrt a reflection through a perpendicular plane. Examples of physical axial vectors are angular velocities and torques. Figure: Axial vector. B. Bona (DAUIN) Vectors and Matrices October 2013 8 / 40

3. Mathematical vectors Mathematical vectors are abstract entities belonging to vector spaces. Vector Space Given a field F = { F ;+ , ·} , a vector space V ( F ), is the set of elements, called vectors , that satisfies the following axiomatic properties: Vector sum : the operation +, called vector sum , is defined so that { V ( F );+ } is a commutative (abelian) group; the identity element is called 0 ; v + 0 = v Product by a scalar : For each α ∈ F and each v ∈ V ( F ), it exists a vector α v ∈ V ( F ); B. Bona (DAUIN) Vectors and Matrices October 2013 9 / 40

Vector space For each α , β ∈ F and each v , w ∈ V ( F ) the following relations hold true: associative property wrt product by a scalar: α ( β v ) = ( αβ ) v existence of the identity wrt product by a scalar: 1( v ) = v ; ∀ v distributive property wrt vector sum: α ( v + w ) = α v + α w distributive property wrt product by a scalar: ( α + β ) v = α v + β v B. Bona (DAUIN) Vectors and Matrices October 2013 10 / 40

Linear functions and dual spaces Given two vector spaces, V ( F ) and U ( F ), both defined on the same field F , a function f : V → U is linear, if, for every v , w ∈ V and λ ∈ F the following axioms are true: f ( v + w ) = f ( v )+ f ( w ) = f v + f w f ( λ v ) = λ f ( v ) = λ f v The linear function L : U → U is also called linear operator , linear transformation , linear application or endomorphism . The set of all linear functions L : U → V defines a linear vector space L ( F ). The set of all linear function L from V ( F ) to F (where usually F = R ) L : V → R , defines a dual vector space, called V ∗ ( F ). Given a vector v ∈ V ( F ), the simplest example of dual vector v ∗ ∈ V ∗ ( F ) is f := v ∗ = v ∗ 1 v 1 + ··· + v ∗ n v n = � v , v ∗ � = � v ∗ , v � ; � v ∗ v ∗ � ··· 1 n B. Bona (DAUIN) Vectors and Matrices October 2013 11 / 40

Scalar product The definition of vector spaces does not include a product between elements of the space. But when a metric (i.e., a measure) is necessary, usually it is that induced by the scalar product. Given two physical vectors a = − QP , b = − → → SR , the scalar product or inner product a · b is a real number defined (geometrically) as: a · b = � a � � b � cos θ where � a � is the vector length and θ , (0 ◦ ≤ θ ≤ 180 ◦ ) is the angle between a and b ; some indicate the product as � a , b � (see also dual spaces ). Given two mathematical vectors a , b ∈ V ( R ) the scalar product is defined as a · b = ∑ a k b k = a T b k B. Bona (DAUIN) Vectors and Matrices October 2013 12 / 40

Properties : distributive wrt sum ( x + y ) · z = x · z + y · z α ( x · y ) = ( α x ) · y = x · ( α y ) distributive wrt product by scalar commutative x · y = y · x positive x · x > 0 , ∀ x � = 0 ; x · x = 0 iff x = 0 Geometrical (physical) definition implies the concepts of angle and length, while in the mathematical setting norm is a quantity derived by the scalar product √ � x � = √ x · x = ∑ � x 2 x T x k = k and the angle between x and y is defined as � x · y � θ = cos − 1 � x � � y � B. Bona (DAUIN) Vectors and Matrices October 2013 13 / 40

Cross product Scalar product · acts between two vectors and produces a scalar; in general we would like to define a product that produces a vector. If we stay in the 3D case, we can define the cross product, aka external product, or vectorial product. Given two physical 3D vectors a = − QP , b = − → → SR , the cross product c = a × b is a vector orthogonal to the plane of a , b , whose length is � c � = � a �� b � sin θ where θ is the minimum angle that takes a to b counterclockwise (right-hand rule). Given two mathematical vectors � T , � T , x , y ∈ R 3 � x 1 � y 1 x = x 2 x 3 y = y 2 y 3 with the cross product x × y is a vector z such that   x 2 y 3 − x 3 y 2 x 3 y 1 − x 1 y 3 z = x × y =   x 1 y 2 − x 2 y 1 B. Bona (DAUIN) Vectors and Matrices October 2013 14 / 40

This relation can be written as       x 2 y 3 − x 3 y 2 0 − x 3 x 2 y 1  =  = S ( x ) y z = x × y = x 3 y 1 − x 1 y 3 x 3 0 − x 1 y 2     x 1 y 2 − x 2 y 1 − x 2 x 1 0 y 3 where S ( x ) is a skew-symmetric (skew-symmetric) matrix. Cross Product Properties : anticommutative x × y = − ( y × x ) distributive wrt sum x × ( y + z ) = ( x × y )+( x × z ) α ( x × y ) = ( α x ) × y = x × ( α y ) distributive wrt product by a scalar non associative x × ( y × z ) � = ( x × y ) × z Jacobi identity a × ( b × c )+ b × ( c × a )+ c × ( a × b ) = 0 B. Bona (DAUIN) Vectors and Matrices October 2013 15 / 40

Given three 3D vectors x , y , z , the triple product is a non associative product: x × ( y × z ) � = ( x × y ) × z with the following (Grassmann) identities: x × ( y × z ) = ( x · z ) y − ( x · y ) z ( x × y ) × z = ( x · z ) y − ( y · z ) x The following relation (between scalars) holds as well ( x × y ) · z = − ( z × y ) · x The cross product is only defined in R 3 ; in order to generalize it in higher dimensional spaces n > 3 it is necessary to introduce the Clifford Algebras. B. Bona (DAUIN) Vectors and Matrices October 2013 16 / 40

Dyadic Product Given two vectors x , y ∈ R n , the dyadic product is defined as ··· x 1 y 1 x 1 y n   . . x ⊗ y = xy T = D ( x , y ) = . . . x i y i .   x n y 1 ··· x n y n Properties ( α x ) ⊗ y x ⊗ ( α y ) = α ( x ⊗ y ) = x ⊗ ( y + z ) = x ⊗ y + x ⊗ z ( x + y ) ⊗ z = x ⊗ y + x ⊗ z ( x ⊗ y ) z = x ( y · z ) x ( y ⊗ z ) = ( x · y ) z B. Bona (DAUIN) Vectors and Matrices October 2013 17 / 40