MATRIX CALCULUS APPLIED TO CRYSTALLOGRAPHY. (short revision) Mois I. Aroyo Universidad del Pais Vasco, Bilbao, Spain

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1 MATRIX CALCULUS APPLIED TO CRYSTALLOGRAPHY (short revision) Mois I. Aroyo Universidad del Pais Vasco, Bilbao, Spain

2 INTRODUCTION TO MATRIX CALCULUS Some of the slides are taken from the presentation Introduction to Matrix Algebra of M. Rademeyer given at the School on Fundamental Crystallography, Bloemfontein, South Africa, 2010

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5 Transposed Matrix! " Let! be a (m( x n) ) matrix The (n( x m) ) matrix obtained from! = (A( ik) ) by exchanging rows and columns is called the transposed matrix! ". Reminder: means

6 Example 1: Transposed Matrix Given that determine! T.

7 Symmetric Matrix A square matrix is symmetric if! " =! i.e. if! =! for "# #" any pair i,k. Symmetric with respect to main diagonal - Top left to bottom right

8 SKEW-SYMMETRIC MATRIX A T =-A If A is a skewsymmetric matrix, then Aii=0, i=1,2,3 as Aik=-Aki

9 EXERCISES Problems 1. Construct the transposed matrix of the (3x1) row matrix: Determine which of the following matrices are symmetric and which are skew-symmetric: A= B= C= D= E= F= (3) G= H= J=

10 Matrix Calculations Multiplication with a number (scalar product): An (m( x n) ) matrix! is multiplied with a number by multiplying each element with :

11 Example 2: Scalar product Given that determine 3!.

12 Matrix addition and subtraction: Let A ik and B ik be general elements of matrices! and ".! and " must be of the same size (i.e. same number of rows and columns). Then the sum and the difference! " is: Element C ik of # is equal to the sum or difference of the elements A ik and B ik of! and " for any pair i,k: C ik = A ik B ik

13 EXERCISES Problems 1. Find 3A-2B, where A= B= Show that the sum of any matrix and its transpose is a symmetric matrix, i.e. (A+A T ) T =A+A T 3. Show that the difference of any matrix and its transpose is a skew-symmetric matrix, i.e. (A-A T ) T =-(A-A T )

14 Matrix multiplication The multiplication of two matrices is only defined when: - the number n (lema) of columns of the left matrix is the same as - the number of m (rima) of rows on the right matrix - no restriction on m (lema) or rows of the left matrix - no restriction on n (rima) or rows of the right matrix # columns of left matrix = # rows of right matrix

15 Multiplication Product of two matrices # and $: The matrix product!"! #$ or is defined by

16 Examples: Matrix Multiplication

17 Example 5: Multiplication Given that and!"#"$%. Determine!. Determine D=BA, check if C=D or not.

18 EXERCISES Problems 1. Find the products AB and BA, if they exist, where 1 2 A= B= Find the matrix products AB and BA of the row vector A= 1 2 3, and the column vector B= Prove that A(BC)=(AB)C where 1 2 A= B= C=

19 Determinants The determinant det(!) ) or! of! can be calculated for any (n( x n) ) square matrix.

20 Determinants

21 Example 6: Determinant Given that Determine det!!".

22 EXERCISES Problems 1. Find the values of the determinants A and B where A= B= Prove that det(ab)=det(a)det(b) where A= B= Prove that det(a)= det(a T ) where A=

23 Inverse of a Matrix A matrix!"which fulfills the condition!# = $ for a square matrix #,, is the inverse matrix # -1 of #%"i.e. ## &'! $( # -1 exists if and only if det(#) 0. Not all matrices have an inverse matrix. Assume that # -1 exists. If!# = $ then #! = $ also holds. A matrix is called orthogonal if A -1 =A T, i.e. AA T =A T A=I

24 EXAMPLE Inverse of a matrix A: (A -1 )ik=(det A) -1 (-1) i+k Bki Find the inverse, if it exists of A, where (i) det A=3, det A 0 (ii) (A -1 )11: (1/3)(-1) 1+1 B11= 11/ B11= det A= = det = 11 (iii) (A -1 )12: (1/3)(-1) 1+2 B21= -9/3... A -1 = 1/ Is it correct?

25 Example 7: Inverse of a Matrix Given that Determine!!".

26 Example 7: Inverse of a Matrix Given that Determine!!". SOLUTION: A -1 = 1/5 0 2/5 2/5 0-1/5 1/15 1/3 2/15

27 Trace of a Matrix The trace of a (n( x n) ) square matrix! is the sum of the elements on the main diagonal.

28 SYMMETRY OPERATIONS AND THEIR MATRIX-COLUMN PRESENTATION

29 Mappings and symmetry operations Definition: A mapping of a set A into a set B is a relation such that for each element a A there is a unique element b B which is assigned to a. The element b is called the image of a. An isometry leaves all distances and angles invariant. An ʻisometry of the first kindʼ, preserving the counter clockwise sequence of the edges ʻshort middle longʼ of the triangle is displayed in the upper mapping. An ʻisometry of the second kindʼ, changing the counter clockwise sequence of the edges of the triangle to a clockwise one is seen in the lower mapping. A parallel shift of the triangle is called a translation. Translations are special isometries. They play a distinguished role in crystallography.

30 Example: Matrix presentation of symmetry operation Mirror symmetry operation Mirror line my at 0,y Matrix representation drawing: M.M. Julian Foundations of Crystallography ctaylor & Francis, 2008 my x = -x y y = -1 1 x y Fixed points my x f yf = x f yf det -1 1 =? tr -1 1 =?

31 Description of isometries coordinate system: {O, a, b, c} isometry: X X ~ (x,y,z) (x,y,z) ~ ~ ~ ~ x = F1(x,y,z)

32 Matrix notation for system of linear equations

33 Matrix-column presentation of isometries linear/matrix part translation column part matrix-column Seitz symbol pair

34 EXERCISES Problem Referred to an orthorhombic coordinated system (a b c; α=β=γ=90) two symmetry operations are represented by the following matrix-column pairs: ( ) ( ) /2 (W1,w1)= 1 0 (W2,w2)= /2 Determine the images Xi of a point X under the symmetry operations (Wi,wi) where Can you guess what is the geometric nature of (W1,w1)? And of (W2,w2)? X= 0,70 0,31 0,95 Hint: A drawing could be rather helpful

35 EXERCISES Problem Characterization of the symmetry operations: det( ) tr( ) =? 1 =? -1-1 What are the fixed points of (W1,w1) and (W2,w2)? ( /2 0 1/2 )xf yf zf = xf yf zf

36 Short-hand notation for the description of isometries (W,w) isometry: X X ~ notation rules: -left-hand side: omitted -coefficients 0, +1, -1 -different rows in one line examples: /2 0 1/2 { -x+1/2, y, -z+1/2 x+1/2, y, z+1/2

37 EXERCISES Problem Construct the matrix-column pair (W,w) of the following coordinate triplets: (1) x,y,z (2) -x,y+1/2,-z+1/2 (3) -x,-y,-z (4) x,-y+1/2, z+1/2

38 Combination of isometries (U,u) X X ~ (W,w) (V,v) X ~

39 EXERCISES Problem Consider the matrix-column pairs of the two symmetry operations: (W1,w1)=( ) ( ) / (W2,w2)= /2 Determine and compare the matrix-column pairs of the combined symmetry operations: (W,w)=(W1,w1)(W2,w2) (W,w) =(W2,w2)(W1,w1) combination of isometries:

40 Inverse isometries X ~ X (W,w) (C,c) X ~ I = 3x3 identity matrix (C,c)(W,w) = (I,o) o = zero translation column (C,c)(W,w) = (CW, Cw+c) C=W -1 c=-cw c=-w -1 w

41 EXERCISES Problem Determine the inverse symmetry operations (W1,w1) -1 and (W2,w2) -1 where (W1,w1)=( ) ( ) / (W2,w2)= /2 Determine the inverse symmetry operation (W,w) -1 (W,w)=(W1,w1)(W2,w2) inverse of isometries:

42 EXERCISES Problem Consider the matrix-column pairs

43 Matrix formalism: 4x4 matrices augmented matrices: point X point X :

44 4x4 matrices: general formulae point X point X : combination and inverse of isometries:

45 EXERCISES Problem Construct the (4x4) matrix-presentation of the following coordinate triplets: (1) x,y,z (2) -x,y+1/2,-z+1/2 (3) -x,-y,-z (4) x,-y+1/2, z+1/2

46 Crystallographic symmetry operations Symmetry operations of an object The isometries which map the object onto itself are called symmetry operations of this object. The symmetry of the object is the set of all its symmetry operations. Crystallographic symmetry operations If the object is a crystal pattern, representing a real crystal, its symmetry operations are called crystallographic symmetry operations. The equilateral triangle allows six symmetry operations: rotations by 120 and 240 around its centre, reflections through the three thick lines intersecting the centre, and the identity operation.

47 EXAMPLE Space group P21/c (No. 14) Geometric interpretation of matrix-column presentation symmetry-elements diagram general-position diagram

48 Crystallographic symmetry operations characteristics: fixed points of isometries geometric elements (W,w)Xf=Xf Types of isometries preserve handedness identity: the whole space fixed translation t: no fixed point x = x + t rotation: one line fixed rotation axis φ = k 360 /N screw rotation: no fixed point screw axis screw vector

49 Crystallographic symmetry operations Screw rotation n-fold rotation followed by a fractional p translation t parallel n to the rotation axis Its application n times results in a translation parallel to the rotation axis

50 Types of isometries do not preserve handedness roto-inversion: centre of roto-inversion fixed roto-inversion axis inversion: centre of inversion fixed reflection: plane fixed reflection/mirror plane glide reflection: no fixed point glide plane glide vector

51 Crystallographic symmetry operations Glide plane reflection followed by a 1 2 fractional translation t parallel to the plane Its application 2 times results in a translation parallel to the plane

52 Matrix-column presentation of some symmetry operations Rotation or roto-inversion around the origin: ( ) W11 W12 W13 W21 W22 W23 W31 W32 W = Translation: ( ) w1 w2 w3 x y z = x+w1 y+w2 z+w3 Inversion through the origin: ( ) -1 0 x -x = -1 0 y -y -1 0 z -z

53 Geometric meaning of (W, w) Type of symmetry operation: W information (a) type of isometry order: W n =I rotation angle

54 EXERCISES Problem Determine the type and order of isometries that are represented by the following matrix-column pairs: (1) x,y,z (2) -x,y+1/2,-z+1/2 (3) -x,-y,-z (4) x,-y+1/2, z+1/2 (a) type of isometry

55 EXERCISES Problem (cont.) Consider the matrix-column pairs Determine the type and order of isometries that are represented by the matrices A, B, C and D:

56 Geometric meaning of (W, w) W information Texto (b) axis or normal direction : (b1) rotations: = (b2) roto-inversions: reflections:

57 EXERCISES Problem Determine the rotation or rotoinversion axes (or normals in case of reflections) of the following symmetry operations (2) -x,y+1/2,-z+1/2 (4) x,-y+1/2, z+1/2 rotations: Y(W) = W k-1 + W k W + I reflections: Y(-W) = - W + I

58 Geometric meaning of (W, w) W information (c) sense of rotation: for rotations or rotoinversions with k>2 det(z): x non-parallel to

59 EXAMPLE Space group P21/c (No. 14) Geometric interpretation of (W,w) -screw or glide components? -fixed points? symmetry-elements diagram general-position diagram

60 Fixed points of isometries (W,w)Xf=Xf solution: point, line, plane or space no solution Examples: solution: ( ) /2 0 x y z = x y z What are the fixed points of this isometry? no solution: ( ) /2 x y z = x y z

61 Geometric meaning of (W,w) (W,w) n =(I,t) (W,w) n =(W n,(w n W+I)w) =(I,t) (A) intrinsic translation part : glide or screw component t/k (A1) screw rotations: = (A2) glide reflections: w

62 Fixed points of (W,w) (B) location (fixed points ): xf (B1) t/k = 0: (B2) t/k 0:

63 EXERCISES Problem Determine the intrinsic translation parts (if relevant) and the fixed points of the following symmetry operations (1) x,y,z (2) -x,y+1/2,-z+1/2 (3) -x,-y,-z (4) x,-y+1/2, z+1/2 screw rotations: Y(W) = W k-1 + W k W + I glide reflections: w fixed points:

64 EXAMPLE Space group P21/c (No. 14) Geometric interpretation of (W,w) -screw or glide components? -fixed points? symmetry-elements diagram general-position diagram

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66 Crystallographic databases Group-subgroup relations Structural utilities Representations of point and space groups Solid-state applications

67 Crystallographic Databases International Tables for Crystallography

68 EXERCISES Problem Construct the matrix-column pairs (W,w) of the following coordinate triplets: (1) x,y,z (2) -x,y+1/2,-z+1/2 (3) -x,-y,-z (4) x,-y+1/2, z+1/2 Characterize geometrically these matrix-column pairs taking into account that they refer to a monoclinic basis with unique axis b.

69 Bilbao Crystallographic Server Problem: Geometric Interpretation of (W,w) SYMMETRY OPERATION

70 ADDITIONAL

71 Multiplication Product of matrix! with column a: Example: How to get element d 1 : d 1 = A 11 a 1 + A 12 a A 1k a k + + A 1n a n

72 Example 3: Multiplication Given that and!"#"$%. Determine!.

73 Multiplication Product of matrix! with row a T : Example: How to get elements d 1 and d 2 : d 1 = a 1 A 11 + a 2 A a k A k1 + + a n A m1 d 2 = a 1 A 12 + a 2 A a k A k2 + + a n A m2

74 Example 4: Multiplication Given that and!"#"$%. Determine!.

75 Using matrix notation for linear equations Consider the set of linear equations: Matrix notation: or or Matrix-column pair or

76 Matrix formalism: Basic results combination of isometries: inverse isometries:

77 Fixed points of (W,w) (W,w) n =(W n,(w n W+I)w) =(I,t) (a1) rotations: = (a2) reflections: =

78 Geometric meaning of (W, w) w -information (B) location (fixed points ): xf (B1) t/k = 0: (B2) t/k 0:

79 Geometric meaning of (W, w) w -information (A) intrinsic translation part : glide or screw component t/k (W,w) n =(W n,(w n W+I)w) =(I,t) (A1) screw rotations: (A2) glide reflections: w

80 Fixed points of isometries (W,w)Xf=Xf solution: point, line, plane or space Example: no solution (W,w)=( ) / /2 ( ) t/k= (W,w) 4 = /2 ( ) (W,w-t/k)= /2 0 What are the fixed points of this isometry?