10/15/12 9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN I Main Topics A MoAvaon B Inverse [A]-1 of a real matrix A C Determinant |A| of a real matrix A D Eigenvectors and eigenvalues (Principal direcAons and principal values) E Diagonalizaon of a matrix F Principal axes theorem (maxima/minima) in 2D G Strain ellipses, strain ellipsoids, and principal strains H Principal values and principal direcAons for symmetric and non-symmetric matrices * Appendices 1 and 2 10/15/12 GG303 1 9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN A forward deformaon deforms a unit circle to a strain ellipse Objecve: To describe size and shape of strain ellipse 10/15/12 GG303 2 1 10/15/12 9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN • For irrotaonal deformaon, the axes which transform to become the principal strain axes (dashed) are not rotated in either the forward deformaon or in the inverse deformaon 10/15/12 GG303 3 9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN • For rotaonal deformaon, the inverse deformaon to retro-deform the strain ellipse back to a unit circle causes lines along the principal strain axes in the strain ellipse to be rotated. • The angular difference between the orientaon of the principal strain axes (solid) and the retro- deformed axes (dashed) in the iniAal state is the rotaon. 10/15/12 GG303 4 2 10/15/12 9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN II MoAvaon A Eigenvectors and eigenvalues provide simple, elegant, and clear ways to solve many scienAfic problems [e.g., geometry, strain, stress, curvature (shapes of surfaces)] B To find the magnitudes and direcAons of the principal strains using linear algebra rather than graphical means or calculus (e.g., Ramsay and Huber, 1984) 10/15/12 GG303 5 9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN -1 ⎡ a b ⎤ III Inverse [A] of a real matrix A [A] = ⎢ ⎥ ⎣ c d ⎦ A [A][A]-1 = [A]-1[A] = I, where I = idenAty matrix B [A] and [A]-1 must be nxn C Inverse [A]-1 of a 2x2 matrix −1 1 ⎡ d −b ⎤ 1 ⎡ d −b ⎤ [A] = ⎢ ⎥ = ⎢ ⎥ ad − bc ⎣ −c a ⎦ A ⎣ −c a ⎦ D Inverse [A]-1 of a 3x3 matrix also requires determinant to be non-zero 10/15/12 GG303 6 3 10/15/12 9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN IV Determinant |A| of a real matrix |A| A A number that provides scaling informaon on a square matrix B Determinant of a 2x2 matrix ⎡ a b ⎤ Akin to: A = ⎢ ⎥, A = ad − bc Cross product (an area) ⎣ c d ⎦ Scalar triple product (a volume) C 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 ⎥ ⎣ ⎦ 10/15/12 GG303 7 9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN IV Determinant (cont.) Intersecting lines have non-parallel normals D Geometric meanings of the real AX = B = 0 matrix equaon AX = B = 0 (1) (1) nx ny x d1=0 1 |A| ≠ 0 ; (2) (2) = nx ny y d2=0 a [A]-1 exists b Describes two lines (or 3 (1) (2) (1) (2) |A| = nx * ny - ny * nx ≠ 0 planes) that intersect at n x n ≠ 0 the origin 1 2 c X has a unique soluAon 2 |A| = 0 ; Parallel lines have parallel normals a [A]-1 does not exist AX = B = 0 (1) (1) b Describes two co-linear nx ny x d1=0 (2) (2) = lines that that pass nx ny y d2=0 through the origin (or three planes that intersect (1) (2) (1) (2) a line or plane through |A| = nx * ny - ny * nx = 0 the origin) n1 x n2 = 0 c X has no unique soluAon 10/15/12 GG303 8 4 10/15/12 9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN V Eigenvectors and eigenvalues A Eigenvalue problems involve the matrix equaon AX = λX 1 A is a (known) square matrix (nxn) 2 X is a non-zero direcAonal eigenvector (nx1) 3 λ is a number, an eigenvalue 4 λX is a vector (nx1) 5 AX is a vector (nx1) 6 Eigenvectors (X) maintain their orientaon when mulAplied by matrix A 7 If X is a unit vector, λ is the length of AX 10/15/12 GG303 9 9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN V Eigenvalue problems, eigenvectors and eigenvalues (cont.) B Eigenvectors have corresponding eigenvalues, and vice-versa C In Matlab, [v,d] = eig(A), finds eigenvectors (v) and eigenvalues (d) 10/15/12 GG303 10 5 10/15/12 9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN V Eigenvalue problems, eigenvectors and eigenvalues (cont.) D Examples ⎡ 1 0 ⎤ ⎡ x ⎤ ⎡ x ⎤ ⎡ x ⎤ = = 1 1 IdenAty matrix (I) ⎢ ⎥ ⎢ y ⎥ ⎢ y ⎥ ⎢ y ⎥ ⎣ 0 1 ⎦ ⎣⎢ ⎦⎥ ⎣⎢ ⎦⎥ ⎣⎢ ⎦⎥ All vectors in the xy-plane maintain their orientaon when operated on by the idenAty matrix, so all vectors are eigenvectors, and all vectors maintain their length, so all eigenvalues for I equal 1 10/15/12 GG303 11 9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN V Eigenvalue problems, eigenvectors and eigenvalues (cont.) D Examples 2 A matrix for rotaons in the xy plane ⎡ cosω cosω ⎤ ⎡ 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 10/15/12 GG303 12 6 10/15/12 9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN V Eigenvalue problems, eigenvectors and eigenvalues (cont.) D Examples 3 A 3D rotaon matrix a The only vectors that are not rotated are along the axis of rotaon b The real eigenvectors of a 3D rotaon matrix give the orientaon of the axis of rotaon c The rotaon does not change the length of vectors, so the real eigenvalues equal 1 10/15/12 GG303 13 9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN V Eigenvalue problems, eigenvectors and eigenvalues (cont.) D Examples ⎡ 0 2 ⎤ Eigenvalue 4 A = ⎢ ⎥ ⎣ 2 0 ⎦ Eigenvector ⎡ 1 ⎤ ⎡ 0 2 ⎤ ⎡ 1 ⎤ ⎡ 2 ⎤ ⎡ 1 ⎤ A ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ = 2 ⎢ ⎥ Eigenvalue ⎣ 1 ⎦ ⎣ 2 0 ⎦ ⎣ 1 ⎦ ⎣ 2 ⎦ ⎣ 1 ⎦ Eigenvector ⎡ 1 ⎤ ⎡ 0 2 ⎤ ⎡ 1 ⎤ ⎡ −2 ⎤ ⎡ 1 ⎤ A ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ = −2 ⎢ ⎥ ⎣ −1 ⎦ ⎣ 2 0 ⎦ ⎣ −1 ⎦ ⎣ 2 ⎦ ⎣ −1 ⎦ 10/15/12 GG303 14 7 10/15/12 9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN V Eigenvalue problems, eigenvectors and eigenvalues (cont.) Eigenvalues D Examples ⎡ 9 3 ⎤ Eigenvectors 5 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 ⎦⎥ 10/15/12 GG303 15 9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN V Eigenvalue problems, eigenvectors and eigenvalues (cont.) E Alternave form of an eigenvalue equaon 1 [A][X]=λ[X] SubtracAng λ[IX] = λ[X] from both sides yields: 2 [A-Iλ][X]=0 (same form as [A][X]=0) F SoluAon condiAons and connecAons with determinants 1 Unique trivial soluAon of [X] = 0 if and only if |A-Iλ|≠0 2 Eigenvector soluAons ([X] ≠ 0) if and only if |A-Iλ|=0 10/15/12 GG303 16 8 10/15/12 9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN V Eigenvalue problems, eigenvectors and eigenvalues (cont.) J CharacterisAc equaon: |A-Iλ|=0 1 The roots of the characterisAc equaon are the eigenvalues 10/15/12 GG303 17 9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN V Eigenvalue problems, eigenvectors and eigenvalues (cont.) J CharacterisAc equaon: |A-Iλ|=0 ⎡ a b ⎤ 2 Eigenvalues of a general 2x2 matrix A = ⎢ ⎥ ⎣ c d ⎦ a − I b a A − Iλ = = 0 c d − λ b (a − λ)(d − λ) − bc = 0 (a+d) = tr(A) (ad-bc) = |A| c λ 2 − (a + d)λ + (ad − bc) = 0 2 (a + d) ± (a + d) − 4(ad − bc) d λ ,λ = 1 2 2 10/15/12 GG303 18 9 10/15/12 9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN V Eigenvalue problems, eigenvectors and eigenvalues (cont.) J CharacterisAc equaon: |A-Iλ|=0 ⎡ a b ⎤ 3 Eigenvalues of a symmetric 2x2 matrix A = ⎢ ⎥ ⎣ b d ⎦ 2 (a + d) ± (a + d) − 4(ad − b2 ) a λ ,λ = 1 2 2 2 (a + d) ± (a + 2ad + d) − 4ad + 4b2 b λ ,λ = 1 2 2 2 a + d ± a − 2ad + d + 4b2 Radical term ( ) ( ) cannot be c λ1,λ2 = 2 negave. 2 (a + d) ± (a − d) + 4b2 Eigenvalues are d λ ,λ = real. 1 2 2 10/15/12 GG303 19 9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN V Eigenvalue problems, eigenvectors and eigenvalues (cont.) K DisAnct eigenvectors (X1, X2) of a symmetric 2x2 matrix are perpendicular (X1 • X2 = 0) 1a AX1 =λ1X1 1b AX2 =λ2X2 AX1 parallels X1, AX2 parallels X2 (property of eigenvectors) Dong AX1 by X2 and AX2 by X1 can test whether X1 and X2 are orthogonal. 2a X2•AX1 = X2•λ1X1 = λ1 (X2•X1) 2b X1•AX2 = X1•λ2X2 = λ2 (X1•X2) 10/15/12 GG303 20 10 10/15/12 9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN K DisAnct eigenvectors (X1, X2) of a symmetric 2x2 matrix are perpendicular (X1 • X2 = 0) (cont.) The leq sides of 2a and 2b are equal ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ x1 ⎡ a b ⎤ x2 x1 ax2 + by2 ⎢ ⎥ • ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ • ⎢ ⎥ y b d y y bx + dy 3a ⎣⎢ 1 ⎦⎥ ⎣ ⎦ ⎣⎢ 2 ⎦⎥ ⎣⎢ 1 ⎦⎥ ⎣⎢ 2 2 ⎦⎥ = ax1x2 + bx1y2 + by1x2 + dy1y2 ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ x2 ⎡ a b ⎤ x1 x2 ax1 + by1 3b ⎢ ⎥ • ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ • ⎢ ⎥ y b d y y bx + dy ⎣⎢ 2 ⎦⎥ ⎣ ⎦ ⎣⎢ 1 ⎦⎥ ⎣⎢ 2 ⎦⎥ ⎣⎢ 1 1 ⎦⎥ = ax1x2 + by1x2 + bx1y2 + dy1y2 10/15/12 GG303 21 9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN K DisAnct eigenvectors (X1, X2) of a symmetric 2x2 matrix are perpendicular (cont.) Since the leq sides of (2a) and (2b) are equal, the right sides must be equal too.