Matrix Equabons of Ellipses and Ellipsoids; Eigenvectors And

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Matrix Equaons of Ellipses and Ellipsoids; Eigenvectors and Eigenvalues GG303, 2014 Lecture 16

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9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN I Main Topics A Elementary linear algebra relaons B Equaons for an ellipse C Equaon of homogeneous deformaon D Eigenvalue/eigenvector equaon

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Examples of 2D homogeneous deformaon ⎡ 2 0 ⎤ F = Note that the symmetry of the displacement ⎢ 0 0.5 ⎥ ﬁelds (or lack thereof) in the examples ⎣ ⎦ corresponds to the symmetry (or lack thereof) in the deformaon gradient matrix [F]. What is a simple way to describe homogeneous deformaon that is geometrically meaningful? What is the geologic relevance?

⎡ 2 1 ⎤ F = ⎢ ⎥ ⎣ 1 2 ⎦

⎡ 1 1 ⎤ F = ⎢ ⎥ ⎣ 1 0 ⎦

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9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN II Elementary linear algebra relaons A Inverse [A]-1 of a real matrix A 1 [A][A]-1 = [A]-1[A] = [I], ⎡ 1 0 ⎤ where [I] = identy matrix (e.g., ) [I ] = ⎢ ⎥ ⎣ 0 1 ⎦ 2 [A] and [A]-1 must be square nxn matrices 3 Inverse [A]-1 of a 2x2 matrix ⎡ a b ⎤ −1 1 ⎡ d −b ⎤ 1 ⎡ d −b ⎤ [A] = ⎢ ⎥ [A] = ⎢ ⎥ = ⎢ ⎥ ⎣ c d ⎦ ad − bc ⎣ −c a ⎦ A ⎣ −c a ⎦ 4 Inverse [A]-1 of a 3x3 matrix also requires determinant |A| to be non-zero

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9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN II Elementary linear algebra relaons B Determinant |A| of a real matrix A 1 A number that provides scaling informaon on a square matrix 2 Determinant of a 2x2 matrix Akin to: ⎡ a b ⎤ A = ⎢ ⎥, A = ad − bc Cross product (an area) ⎣ c d ⎦ Scalar triple product (a volume) 3 Determinant of a 3x3 matrix

⎡ a b c ⎤ ⎢ ⎥ e f d f d e A = d e f , A = a − b + c ⎢ ⎥ h i g i g h ⎢ g h i ⎥ ⎣ ⎦

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9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN II Elementary linear algebra relaons

⎡ a b ⎤ T ⎡ a c ⎤ C Transpose For [A] = ⎢ ⎥, [A] = ⎢ ⎥ ⎣ c d ⎦ ⎣ b d ⎦ D A matrix [A] is symmetric if [A] = [A]T E Transpose of a matrix product

⎡ a b ⎤ ⎡ e f ⎤ T ⎡ a c ⎤ T ⎡ e g ⎤ If [A] = and [B] = ⎢ ⎥, then [A] = and [B] = ⎢ ⎥ ⎢ c d ⎥ g h ⎢ b d ⎥ f h ⎣ ⎦ ⎣⎢ ⎦⎥ ⎣ ⎦ ⎣⎢ ⎦⎥

⎡ ae + bg af + bh ⎤ T ⎡ ae + bg ce + dg ⎤ [A][B] = ⎢ ⎥, ⎡[A][B]⎤ = ⎢ ⎥ ce + dg cf + dh ⎣ ⎦ af + bh cf + dh ⎣⎢ ⎦⎥ ⎣⎢ ⎦⎥

T T ⎡ ea + gb ec + gd ⎤ T [B] [A] = ⎢ ⎥ = ⎡[A][B]⎤ fa + hb fc + hd ⎣ ⎦ This is true for any real nxn matrices ⎣⎢ ⎦⎥

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9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN II Elementary linear algebra relaons F Representaon of a dot product using matrix mulplicaon and the matrix transpose ! ! a • b = ax ,ay ,az • bx ,by ,bz = axbx + ayby + azbz ⎡ b ⎤ ⎢ x ⎥ ⎡ ⎤ T = ax ay az ⎢ by ⎥ = [a] [b] ⎣ ⎦⎢ ⎥ bz ⎣⎢ ⎦⎥

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III Equaons for an ellipse A Equaon of a unit circle ! ! 1 x2 + y2 = X • X = 1

⎡ x ⎤ T 2 ⎡ x y ⎤ = X X = 1 ⎣ ⎦⎢ y ⎥ [ ] [ ] ⎣⎢ ⎦⎥ 3 x = cosθ y = sinθ

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III Equaons for an ellipse B Ellipse centered at (0,0), aligned along x,y axes 1 Standard form 2 2 ⎛ x⎞ ⎛ y⎞ + = 1 ⎝⎜ a⎠⎟ ⎝⎜ b⎠⎟ 2 General form Ax2 + Dy2 + F = 0 3 Matrix form

⎡ A 0 ⎤⎡ x ⎤ ⎡ Ax ⎤ A, D, and F are ⎡ x y ⎤ = ⎡ x y ⎤ = −F ⎣ ⎦⎢ ⎥⎢ y ⎥ ⎣ ⎦⎢ Dy ⎥ ⎣ 0 D ⎦⎣⎢ ⎦⎥ ⎣⎢ ⎦⎥ constants here, not matrices

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9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN

III Equaons for an ellipse B Ellipse centered at (0,0), aligned along x,y axes 4 Parametric form x = acosθ y = bsinθ 5 Vector form ! ! ! r = acosθ i + bsinθ j

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III Equaons for an ellipse C Ellipse centered at (0,0), arbitrary orientaon 1 General form Ax2 + (B + C)xy + Dx2 + F = 0 provided 4AD > (B+C)2 2 Matrix form

⎡ A B ⎤⎡ x ⎤ ⎡ Ax + By ⎤ A, B, C, D, and F ⎡ x y ⎤⎢ ⎥⎢ ⎥ = ⎡ x y ⎤⎢ ⎥ = −F ⎣ ⎦ C D y ⎣ ⎦ Cx + Dy are constants here, ⎣ ⎦⎣⎢ ⎦⎥ ⎣⎢ ⎦⎥ not matrices

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9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN IV Equaon of homogeneous deformaon A [X’] = [F][X] B 2D ⎡ ∂x′ ∂x′ ⎤ ⎢ ⎥ ⎡ ⎤ ⎡ dx′ ⎤ ∂x ∂y ⎡ dx ⎤ ⎡ x′ ⎤ ⎡ a b ⎤⎡ x ⎤ Fx′x Fx′y ⎡ x ⎤ = ⎢ ⎥ ⇒ = = ⎢ ⎥ ⎢ dy ⎥ ⎢ ⎥⎢ dy ⎥ ⎢ y ⎥ ⎢ ⎥⎢ y ⎥ ⎢ y ⎥ ⎢ ′ ⎥ ∂y′ ∂y′ ⎢ ⎥ ⎢ ′ ⎥ c d ⎢ ⎥ ⎢ Fy′x Fy′y ⎥⎢ ⎥ ⎣ ⎦ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦⎣ ⎦ ∂x ∂y ⎣⎢ ⎦⎥ C 3D For homogeneous ⎡ ∂x′ ∂x′ ∂x′ ⎤ ⎢ ⎥ strain, the ⎢ ∂x ∂y ∂z ⎥ ⎡ F F F ⎤ ⎡ dx′ ⎤ ⎡ dx ⎤ ⎡ x′ ⎤ x′x x′y x′z ⎡ x ⎤ derivaves are ⎢ ∂y′ ∂y′ ∂y′ ⎥ ⎢ ⎥ ⎢ dy ⎥ ⎢ dy ⎥ ⎢ y ⎥ F F F ⎢ y ⎥ uniform ⎢ ′ ⎥ = ⎢ ⎥⎢ ⎥ ⇒ ⎢ ′ ⎥ = ⎢ y′x y′y y′z ⎥⎢ ⎥ ⎢ ∂x ∂y ∂z ⎥ ⎢ ⎥ ⎢ dz′ ⎥ ⎢ dz ⎥ ⎢ z′ ⎥ F F F ⎢ z ⎥ (constants) , and ⎣ ⎦ ⎢ ∂z′ ∂z′ ∂z′ ⎥⎣ ⎦ ⎣ ⎦ ⎣⎢ z′x z′y z′z ⎦⎥⎣ ⎦ ⎢ ⎥ dx, dy can be ∂x ∂y ∂z ⎣⎢ ⎦⎥ small or large

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IV Equaon of F Proof X T |X|=1 homogeneous [X] [X] = 1 deformaon [X’] = [F][X] [X′] = [F][X] Now solve for [X] −1 −1 D Crical maer: [F] [X′] = [F] [F][X] = [I ][X] = [X] Understanding the −1 [X] = [F] [X′] Now solve for [X]T geometry of the T T X T ⎡ F −1 X ⎤ X T ⎡ F −1 ⎤ deformaon [ ] = ⎣[ ] [ ′]⎦ = [ ′] ⎣[ ] ⎦ E In homogeneous Now substute for [X]T and [X] in ﬁrst equaon

deformaon, a unit T T −1 T −1 X X X ⎡ F ⎤ F X 1 circle transforms to [ ] [ ] = [ ′] ⎣[ ] ⎦ [ ] [ ′] = T an ellipse (and a [X′] [Symmetric matrix][X′] = 1 sphere to an Equaon of ellipse ellipsoid) See slide 11 10/11/14 GG303 13

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IV Equaon of homogeneous deformaon [X’] = [F][X] G [F] transforms a unit circle to a “strain ellipse” H “Strain ellipse” geometrically represents [F][X] I [F]-1 transforms a “strain ellipse” back to a unit circle J [F]-1 transforms a unit circle to a “reciprocal strain ellipse” K [F] transforms a “reciprocal strain ellipse” back to a unit circle L “Reciprocal strain” ellipse geometrically represents [F]-1[X]

⎡ a b ⎤ −1 1 ⎡ d −b ⎤ [F] = ⎢ ⎥ [F] = ⎢ ⎥ ⎣ c d ⎦ ad − bc ⎣ −c a ⎦

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V Eigenvectors and eigenvalues A The eigenvalue matrix equaon [A][X] = λ[X] 1 [A] is a (known) square matrix (nxn) 2 [X] is a non-zero direconal eigenvector (nx1) 3 λ is a number, an eigenvalue 4 λ[X] is a vector (nx1) parallel to [X] 5 [A][X] is a vector (nx1) parallel to [X]

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A The eigenvalue matrix equaon [A][X] = λ[X] (cont.) 6 The vectors [[A][X]], λ[X], and [X] share the same direcon if [X] is an eigenvector 7 If [X] is a unit vector, λ is the length of [A][X]

8 Eigenvectors [Xi] have corresponding eigenvalues [λi], and vice-versa 9 In Matlab, [vec,val] = eig(A), ﬁnds eigenvectors (vec) and eigenvalues (val)

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V Eigenvectors and eigenvalues (cont.) B Examples 1 Identy matrix [I] ⎡ 1 0 ⎤ ⎡ x ⎤ ⎡ x ⎤ ⎡ x ⎤ = = 1 ⎢ ⎥ ⎢ y ⎥ ⎢ y ⎥ ⎢ y ⎥ ⎣ 0 1 ⎦ ⎣⎢ ⎦⎥ ⎣⎢ ⎦⎥ ⎣⎢ ⎦⎥ All vectors in the xy-plane maintain their orientaon when operated on by the identy matrix, so all vectors are eigenvectors, and all vectors maintain their length, so all eigenvalues of [I] equal 1. The eigenvectors are not uniquely determined but could be chosen to be perpendicular.

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III Equaons for an ellipse B Examples (cont.) 2 Ellipse centered at (0,0), aligned along x,y axes

⎡ x′ ⎤ ⎡ ⎤⎡ x ⎤ = 2 0 ⎢ y ⎥ ⎢ ⎥⎢ y ⎥ ⎣⎢ ′ ⎦⎥ ⎣ 0 1.5 ⎦⎣⎢ ⎦⎥ Eigenvectors Eigenvalues [X’] = [F][X]

⎡ 2 0 ⎤⎡ 1 ⎤ ⎡ 2 ⎤ ⎡ 1 ⎤ Note that [F] is symmetric ⎢ ⎥⎢ ⎥ = ⎢ ⎥ = 2 ⎢ ⎥ ⎣ 0 1.5 ⎦⎣⎢ 0 ⎦⎥ ⎣⎢ 0 ⎦⎥ ⎣⎢ 0 ⎦⎥

Unit vectors along the x- [F] [X1 ] = λ1 [X1 ]

and y- axes don’t rotate ⎡ 2 0 ⎤⎡ 0 ⎤ ⎡ 0 ⎤ ⎡ 0 ⎤ ⎢ ⎥⎢ ⎥ = ⎢ ⎥ = 0.5 ⎢ ⎥ and hence are ⎣ 0 1.5 ⎦⎣⎢ 1 ⎦⎥ ⎣⎢ 0.5 ⎦⎥ ⎣⎢ 1 ⎦⎥ F X = λ X eigenvectors [ ] [ 2 ] 2 [ 2 ]

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V Eigenvectors and eigenvalues (cont.) B Examples (cont.) 3 A matrix for rotaons in the xy plane

⎡ cosω sinω ⎤⎡ x ⎤ ⎡ x ⎤ = λ ⎢ ⎥⎢ y ⎥ ⎢ y ⎥ ⎣ −sinω cosω ⎦⎣⎢ ⎦⎥ ⎣⎢ ⎦⎥ All non-zero real vectors rotate; a 2D rotaon matrix has no real eigenvectors and hence no real eigenvalues

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V Eigenvectors and eigenvalues (cont.) B Examples (cont.) 4 A 3D rotaon matrix a The only unit vector that is not rotated is along the axis of rotaon b The real eigenvector of a 3D rotaon matrix gives the orientaon of the axis of rotaon c A rotaon does not change the length of vectors, so the real eigenvalue equals 1

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V Eigenvectors and eigenvalues (cont.) B Examples (cont.) Eigenvalues ⎡ 0 2 ⎤ 5 A = ⎢ ⎥ Eigenvectors ⎣ 2 0 ⎦ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 2 2 ⎡ 0 2 ⎤ 2 2 ⎡ 2 ⎤ 2 2 A ⎢ ⎥ = ⎢ ⎥⎢ ⎥ = ⎢ ⎥ = 2 ⎢ ⎥ ⎢ 2 2 ⎥ 2 0 ⎢ 2 2 ⎥ ⎢ 2 ⎥ ⎢ 2 2 ⎥ ⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦ ⎣ ⎦

⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 2 2 ⎡ 0 2 ⎤ 2 2 ⎡ − 2 ⎤ 2 2 A ⎢ ⎥ = ⎢ ⎥⎢ ⎥ = ⎢ ⎥ = −2 ⎢ ⎥ ⎢ − 2 2 ⎥ 2 0 ⎢ − 2 2 ⎥ ⎢ 2 ⎥ ⎢ − 2 2 ⎥ ⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦ ⎣ ⎦

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V Eigenvectors and eigenvalues (cont.) B Examples (cont.) Eigenvalues 6 ⎡ 9 3 ⎤ Eigenvectors A = ⎢ ⎥ ⎣ 3 1 ⎦

⎡ −3 0.1 ⎤ ⎡ 9 3 ⎤ ⎡ −3 0.1 ⎤ ⎡ −30 0.1 ⎤ ⎡ −3 0.1 ⎤ A ⎢ ⎥ = ⎢ ⎥ = ⎢ ⎥ = 10 ⎢ ⎥ ⎢ 3 1 ⎥ ⎣⎢ − 0.1 ⎦⎥ ⎣ ⎦ ⎣⎢ − 0.1 ⎦⎥ ⎣⎢ −10 0.1 ⎦⎥ ⎣⎢ − 0.1 ⎦⎥

⎡ 0.1 ⎤ ⎡ 9 3 ⎤ ⎡ 0.1 ⎤ ⎡ 0 ⎤ ⎡ 0.1 ⎤ A ⎢ ⎥ = ⎢ ⎥ = = 0 ⎢ ⎥ ⎢ 3 1 ⎥ ⎢ 0 ⎥ ⎣⎢ −3 0.1 ⎦⎥ ⎣ ⎦ ⎣⎢ −3 0.1 ⎦⎥ ⎣ ⎦ ⎣⎢ −3 0.1 ⎦⎥

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V Eigenvectors and eigenvalues (cont.) C We will use eigenvectors and eigenvalues to ﬁnd the axes of all strain ellipses and strain ellipsoids in Lab 9.

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