Sect. 4.2: Orthogonal Transformations For convenience, change of notation: x  x1, y  x2, z  x3, x´  x1, y´  x2, z´  x3 Also: aij  cosθij In new notation, transformation eqtns between primed & unprimed coords become: x1 = a11 x1+a12 x2 +a13 x3 x2 = a21 x1+a22 x2 +a23 x3 x3 = a31 x1+a32 x2 +a33 x3 Or: xi = ∑j aij xj (i,j = 1,2,3) (1) (1) = An example of what mathematicians call a Linear (or Vector) Transformation.

 Einstein summation convention For convenience, another change of notation: If the index is repeated, summation over it is implied.  xi = ∑j aij xj (i,j = 1,2,3)   xi = aij xj (i,j = 1,2,3)  Einstein summation convention To avoid possible ambiguity when powers of an indexed quantity occur: ∑i(xi)2  xixi For the rest of the course, summation convention is automatically assumed, unless stated otherwise.

Linear Transformation: xi = aij xj (i,j = 1,2,3) (1) With aij  cosθij as derived, (1) is only a special case of a general linear transformation, since, as already discussed, the direction cosines cosθij are not all independent. Re-derive connections between them, use new notation. Both coord systems are Cartesian:  Square of magnitude of vector = sum of squares of components. Magnitude is invariant on transformation of coords:  xixi = xixi Using (1), this becomes: aijaikxjxk = xixi (i,j,k = 1,2,3)

aijaikxjxk = xixi (i,j,k = 1,2,3) aijaik = δj,k (j,k = 1,2,3) Can be valid if & only if aijaik = δj,k (j,k = 1,2,3) Identical previous results for orthogonality of direction cosines. Any Linear Transformation: xi = aij xj (i,j = 1,2,3) (1)  Orthogonal Transformation aijaik = δj,k  Orthogonality Condition

Linear (or Vector) Transformation. xi = aijxj (i,j = 1,2,3) (1) Can arrange direction cosines into a square matrix: a11 a12 a13 A  a21 a22 a23 a31 a32 a33 Consider coordinate axes as column vector components: x1 x1 r = x2 r = x2 x3 x2  Coordinate transformation reln can be written: r = Ar with A  Transformation matrix or rotation matrix (or tensor)

Example: 2d Coordinate Rotation Application to 2d rotation. See figure: Easy to show that: x3 = x3 x1 = x1cos + x2sin = x1cos + x2cos( - π/2) x2 = -x1sin  + x2cos = x1cos( + π/2) + x2cos

 Transformation matrix has form: a11 a12 0 cos sin 0 2d rotation. See fig:  aij  cosθij a33 = cosθ33 = 1 a11 = cosθ11 = cos a22 = cosθ22 = cos a12 = cosθ12 = cos( - π/2) = sin a21 = cosθ21 = cos( + π/2) = -sin a31 = cosθ31 = cos(π/2) = 0, a32 = cosθ32 = cos(π/2) = 0  Transformation matrix has form: a11 a12 0 cos sin 0 A = a21 a22 0 = -sin cos 0 0 0 1 0 0 1

 Need only one angle to specify a 2d rotation. 2d rotation. See fig:  aij  cosθij Orthogonality Condition: aijaik = δj,k  a11a11 + a21a21 = 1 a12a12 + a22a22 = 1 , a11a12 + a21a22 = 0 Use expressions for aij & get: cos2 + sin2 =1 sin2 + cos2 =1, cossin - sincos = 0  Need only one angle to specify a 2d rotation.

Transformation matrix A  Math operator that, acting on unprimed system, transforms it to primed system. Symbolically: r = Ar (1)  Matrix A, acting on components of the r in unprimed system yields components of r in the primed system. Assumption: Vector r itself is unchanged (in length & direction) on operation with A. (r2 = (r)2) NOTE: Same formal mathematics results from another interpretation of (1): A acts on r & changes it into r . Components of 2 vectors related by (1). Which interpretation depends on context of problem. Usually, for rigid body motion, use 1st interpretation. For general transformation (1), nature of A depends on which interpretation is used. A acting on coords: Passive transformation. A acting on vector: Active transformation

Example from Marion In the unprimed system, point P is represented as (x1, x2, x3) = (2,1,3). In the primed system, x2 has been rotated from x2, towards x3 by a 30º angle as in the figure. Find the rotation matrix A & the representation of P = (x1, x2, x3) in the primed system.

From figure, using aij  cosθij a11 = cosθ11 = cos(0º) =1 a12 = cosθ12 = cos(90º) = 0 a13 = cosθ13 = cos(90º) = 0 a21 = cosθ21 = cos(90º) = 0 a22 = cosθ22 = cos(30º) = 0.866 a23 = cosθ23 = cos(90º-30º) = cos(60º) = 0.5 a31 = cosθ31 = cos(90º) = 0 a32 = cosθ32 = cos(90º+30º) = -0.5 a33 = cosθ33 = cos(30º) = 0.866 1 0 0  A = 0 0.866 0.5 0 -0.5 0.866

To find new representation of P, apply r = Ar or x1 = a11 x1+a12 x2 +a13 x3 x2 = a21 x1+a22 x2 +a23 x3 x3 = a31 x1+a32 x2 +a33 x3 Using (x1, x2, x3) = (2,1,3)  x1 = x1 = 2 x2 = 0.866x2 +0.5x3 = 2.37 x3 = -0.5 x2 + 0.866x3 = 2.10  (x1, x2, x3) = (2,2.37,2.10)

Useful Relations  cos2α + cos2β + cos2γ = 1 Consider a general line segment, as in the figure: Angles α, β, γ between the segment & x1, x2, x3  Direction cosines of line  cosα, cosβ, cosγ Manipulation, using orthogonality relns from before:  cos2α + cos2β + cos2γ = 1

cosα, cosβ, cosγ, & cosα , cos β, cosγ Angle θ between segments: Consider 2 line segments, direction cosines, as in the figure: cosα, cosβ, cosγ, & cosα , cos β, cosγ Angle θ between segments: Manipulation (trig):  cosθ = cosαcosα +cosβcosβ +cosγcosγ

Sect. 4.3: Formal (math) Properties of the Transformation Matrix For a while (almost) pure math! 2 successive orthogonal transformations B and A, acting on unprimed coordinates: r = Br followed by r = Ar = ABr In component form, application of B followed by A gives (summation convention assumed, of course!): xk = bkjxj , xi = aikxk = aikbkjxj (1) (i,j,k = 1,2,3) Rewrite (1) as: xi = cijxj (2) (2) has the form of an orthogonal transformation C  AB with elements of the square matrix C given by cij  aikbkj

Products  Product of 2 orthogonal transformations B (matrix elements bkj) & A (matrix elements aik) is another orthogonal transformation C = AB (matrix elements cij  aikbkj). Proof that C is also orthogonal: See Prob. 1, p 180. Can show (student exercise!): Product of orthogonal transformations is not commutative: BA  AB Define: D  BA (matrix elements dij  bikakj). Find, in general: dij  cij.  Final coords depend on order of application of A & B. Can also show (student exercise!): Products of such transformations are associative: (AB)C = A(BC)

Note: Text now begins to use vector r & vector x interchangeably Note: Text now begins to use vector r & vector x interchangeably!  r = Ar  x = Ax can be represented in terms of matrices, with coord vectors being column vectors: x = Ax  xi = aijxj or: x1 a11 a12 a13 x1 x2 = a21 a22 a23 x2 x2 a31 a32 a33 x3 Addition of 2 transformation matrices: C = A + B  Matrix elements are: cij = aij + bij

Inverse Define the inverse A-1 of transformation A: x = Ax (1),  x  A-1 x (2) In terms of matrix elements, these are: xk = akixi (1 ), xi  aij xj (2) where aij are matrix elements of A-1 Combining (1) & (2):  xk = akiaij xj clearly, this can hold if & only if: akiaij = δj,k (3) Define: Unit Matrix 1 0 0 1  0 1 0  akiaij = δj,k are clearly matrix 0 0 1 elements of 1

Transpose  In terms of matrices, DEFINE A-1 by: AA-1  A-1A  1 Proof that AA-1  A-1A : p 146 of text. 1  Identity transformation because: x = 1 x and A = 1 A Matrix elements of A-1 & of A are related by: aij = aji (4) Proof of this: p 146-147 of text. Define: Ã  Transpose of A  matrix obtained from A by interchanging rows & columns. Clearly, (4)  A-1 = Ã & thus: ÃA = AÃ = 1

Combine aij = aji with akiaij = δj,k  akiaji = δj,k (5) A-1 = Ã  For orthogonal matrices, the reciprocal is equal to the transpose. Combine aij = aji with akiaij = δj,k  akiaji = δj,k (5) (5): A restatement of the orthogonality relns for the aki ! Dimension of rectangular matrix, m rows, n columns  m  n. A, A-1, Ã : Square matrices with m = n. Column vector (1 column matrix) x, dimension m  1. Transpose x: dimension 1  m (one row matrix). Matrix multiplication: Product AB exists only if # columns of A = # rows of B: cij = aikbkj See text about multiplication of x & its transpose with A & Ã

Obviously, diagonal elements in this case: aii = 0 Define: Symmetric Matrix  A square matrix that is the same as its transpose: A = Ã  aij = aji Antisymmetric Matrix  A square matrix that is the negative of its transpose: A = - Ã  aij = - aji Obviously, diagonal elements in this case: aii = 0

2 interpretations of orthogonal transformation Ax = x 1) Transforming coords. 2) Transforming vector x. How does arbitrary vector F (column matrix) transform under transformation A? Obviously, G  AF (some other vector). If also, the coord system is transformed under operation B, components of G in new system are given by G  BG  BAF Rewrite (using B-1B = 1) as: G = BG = BAB-1BF Also, components of F in new system are given by F  BF

Combining gives: G = BAB-1F where: F  BF, G  BG  If define operator BAB-1  A we have: G  A F (same form as G = AF, but expressed in transformed coords)  Transformation of operator A under coord transformation B is given as: A  BAB-1  Similarity transformation

Some identities (no proofs): |AB| = |A||B| Properties of determinant formed from elements of an orthogonal transformation matrix: det(A)  |A| Some identities (no proofs): |AB| = |A||B| From orthogonality reln ÃA = AÃ = 1 get |Ã||A| = |A||Ã| = 1 Determinant is unaffected by interchange of rows & columns: |Ã| = |A| Using this with above gives: |A|2 = 1  |A| =  1

Multiply from right by B: AB = BAB-1B = BA Value of determinant is invariant under a similarity transformation. Proof: A, B orthogonal transformations Assumes 1) B-1 exists & 2) |B|  0 Similarity transformation: A  BAB-1 Multiply from right by B: AB = BAB-1B = BA Determinant: |A||B| = |B||A| (|B| = a number  0)  Divide by |B| on both sides & get |A| = |A|