2. 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.

3.  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.

4. 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)

5. 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

6. 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:
x x1
r = x2 r = x2
x x2
 Coordinate transformation reln can be written:
r = Ar with A  Transformation matrix or rotation matrix (or tensor)

7. 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

8.  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 a cos sin 0
A = a21 a = sin cos 0

9.  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.

10. 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

11. 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.

12. 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
 A =

13. 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 x x3 = 2.10
 (x1, x2, x3) = (2,2.37,2.10)

14. 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

15. 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

16. 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

17. 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)

18. 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 a x1
x2 = a21 a22 a x2
x2 a31 a32 a x3
Addition of 2 transformation matrices: C = A + B  Matrix elements are: cij = aij + bij

19. 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   akiaij = δj,k are clearly matrix elements of 1

20. 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 of text.
Define: Ã  Transpose of A  matrix obtained from A by interchanging rows & columns.
Clearly, (4)  A-1 = Ã & thus: ÃA = AÃ = 1

21. 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

22. 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

23. 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

24. 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

25. 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

26. 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|