Vectors and Matrices

Transcription

1 Chapter Vectors and Matrices. Introduction Vectors and matrices are used extensively throughout this text. Both are essential as one cannot derive and analyze laws of physics and physical measurements without vectors and one cannot process these measurements in a digital computer without matrices. Further, each has powerful features that when comined can result in considerale derivation simplifications. We shall highlight the main similarities and the sutle differences etween them. In this context we will e concerned with Euclidean vector spaces for which the inner product is defined. In n-dimensional Euclidean vector spaces it is possile to construct a set of n orthogonal unit vectors r, r,, r n. As such, an aritrary vector v in this space is represented y v = v + r + vr + v n rn (.) where v,v,,vn are the scalar coordinates of v. In the special case of three dimensional vector space, the unit vectors r, r, r can e graphically represented y a set of three orthogonal axes. Hence, v,v,v (the coordinates of the -dimensional vector v) are the projections of v along { r, r, r}. A matrix on the other hand, is an array of n rows and m columns, of n m numers [,]. For example, a matrix A is represented y 7

2 8 Introduction to Modern Navigation Systems a A a a a a a a a a The transpose of a i j matrix B is a j i matrix, denoted y B, in which the rows and columns of B trade places. For example, the transpose of the aove matrix A is given y a a a A a a a a a a In particular, a n array is a column matrix and a n array is a row matrix. Transposing a row matrix makes it a column matrix and vice versa. To adapt vectors to matrix manipulations it is common to use row or column matrices whose components are the vector coordinates. When there is no confusion, column matrices will e used to denote vectors. So vector v, in Eq. (.), can e represented in matrix form y v v v vn (.) Implicit in the aove equation that the components v,v,,vn are the coordinates of v along the vectors r, r,, r n.

3 . Vector Inner Product Vectors and Matrices 9 The inner product of the two real vectors u and v is defined y [,4] u v u v u v u n v n (.) where { u,u,,un} and { v,v,,vn} are the coordinates of u and v. The length of vector u, also called its norm, is defined y the inner product norm( u ) u u u u u un (.4) We call u and v orthogonal if their inner product equals zero; if, in addition, each is of unit length then they are orthonormal. In particular the set of vectors r, r,, r n, introduced aove, are orthonormal ecause r i ri and ri rj, i, j,,,n; i j (.5) Thus it can e seen from Eqs. (.) and (.5) that the coordinates of vector v are given y v = v r, i,,,n (.6) i i In a three dimensional space the inner product has a physical significance. The cosine of the angle etween vectors u and v, denoted y cos(u, v) is determined y the cosine law (see Appendix A) u v cos ( u, v) (.7) u v

4 Introduction to Modern Navigation Systems If v is a unit vector we get from Eqs. (.6) and (.7) vi vr =cos( v, r ), i,,,n i i (.8) When a vector set { r, r,, rn} is represented in column matrices, then they are orthonormal if rr i i and rr i j, i, j,,,n; i j (.9) The Hermitian inner product of the complex vectors u and v is defined y uv uv uv unvn uv (.) where the prime denotes the matrix transpose and the * denotes the complex conjugate. A matrix is called square if its numer of rows is the same as its numer of columns. The identity matrix is a real square whose diagonal components are all ones and the rest are all zeros. For example the identity matrix is I. Vector Cross Products and Skew Symmetric Matrix Algera Suppose that { r, r, r} is an orthonormal vector set for a - dimensional vector space in which the two vectors u and v are given y

5 Vectors and Matrices u v u r v r u v r r u v r r then the vector cross product of u and v, denoted y w u v, is defined y [5,6] w ( uv uv) r ( uvuv) r ( uv uv) r (.) Notice that the cross product of the two vectors u and v is another vector that is orthogonal to oth u and v. In matrix notation w is given y uv uv w uv uv uv uv uv It is shown in Appendix A that if u and v are unit vectors then the magnitude of their vector cross product, w, is the sine of the angle etween them. Therefore if k is the unit vector along w then w u v sin( u, v) k (.) A skew symmetric matrix B is a matrix with the property of B = -B. A -dimensional skew symmetric matrix emulates the cross product operation and enales it to e expressed in matrix notation. The vector corresponds to the skew symmetric matrix defined y

6 Introduction to Modern Navigation Systems S( ) (.) The operator S and the tilde notation are identical and will e used interchangealy to denote skew symmetric matrices, even though the latter will e used whenever possile. Properties of the Skew Symmetric Matrix In the following it is assumed that a, and w are three dimensional aritrary vectors (column matrix) and and are aritrary scalars.. Correspondence to vector cross product: w w w w w w w w (.4) w w w Thus operating the matrix product of ~ and w corresponds to the vector cross product of the and w.. Linearity: a a S( a) S( ) a a a a a a S( a) S( ) a a a a

7 Vectors and Matrices S( a) S( ) S( a ) (.5). Skewness ~ ' ' ~ Thus ~ ' ~ (.6) 4. Operating on self vector (.7) Lemma. w w ~~ w ~ ~ S( ) (.8) Proof: Since w w w w w w w w w w w w w w w w w w w (.9)

8 4 Introduction to Modern Navigation Systems The RHS matrix can e arranged in the form of w ( w) I w (.) Likewise w ( w) I w (.) Sutracting Eq. (.) from Eq. (.) gives www w w w w w w w w w (.) w w w w w w S w w S( w) w w Before we explore further properties of the skew matrix we now introduce the orthonormal matrix. A matrix R r r... r n is called orthonormal if its columns are mutually orthonomal, or equivalently satisfies Eq. (.9). A three-dimensional orthonormal matrix C c c c (.) have these properties

9 Vectors and Matrices 5 c c c c c c c c c,, (.4) Lemma. If C is orthonormal matrix, then CS( ) C S( C) (.5) Proof: Since CC c c c C Multiplying C y ~ and expanding C yields ' c ' ' c CC c c c c c c c Multiplying the two right hand side matrices and collecting terms gives CC c c c c c c c c c c c c ' ' ' ' ' ' ( ) ( ) ( ) Applying Eq. (.) to the aove equation yields CC c c cc cc S( ) S( ) S( )

10 6 Introduction to Modern Navigation Systems Sustituting from Eq. (.4) in the aove equation gives CC c c c S( ) S( ) S( ) By virtue of the linearity of the S operator it follows that CC S( c c c)=s( c c c ) S( C) In this chapter we explored the little differences etween the vector and matrix notations. In general, a vector represents a direction in the three dimensional Euclidean space and a magnitude. A matrix is a set of elements that are arranged in a specific manner. A vector cross product is a special operation pertains to vectors ut can e emulated in matrix notations using the skew symmetric matrix. One advantage with matrix products is their associativity, ( AB) C A( BC) which is not the case for vector cross products, ( a) c a( c ). Awareness of these distinctions will allow us to move from one notation to the other as desired. In the following chapter, the usefulness of these tools will e very vivid as they allow us to descrie vector rotations (that are given in vector notations) in terms of transformation matrices. References. D. E. Bourne, P. C. Kendall, Vector Analysis, Allyn and Bacon Inc., Boston, MA, R. Bellman, Introduction to Matrix Analysis, McGraw Hill, New York, New York, 97.. R. Larson, R. Hostetler, B. Edwards, Calculus with Analytic Geometry, D. C. Heath and Company, Lexington, Ma, D. Varerg, E. Purcell, Calculus with Analytic Geometry, Prentice Hall, Englewood Cliffs, New Jersey, 99.