Fakultät für Physik R: Rechenmethoden für Physiker, WiSe 015/16 Dozent: Jan von Delft Übungen: Benedikt Bruognolo, Dennis Schimmel, Frauke Schwarz, Lukas Weidinger http://homepages.physik.uni-muenchen.de/~vondelft/lehre/15r/ Sheet 07.5: Matrices II: Inverse, Basis Transformation Posted: Friday, 7.11.15 Due: Friday, 04.1.15, 13:00 Central Tutorial: 09.1.15 [](E/M/A) means: problem counts points and is easy/medium hard/advanced Example Problem 1: Calculating determinants [] Points: [](E) Compute the determinants of the following matrices by expanding them along an arbitrary row or column. Hint: The more zeros it contains, the easier the calculation. ( ) ( ) a a a 0 3 1 1 A =, B = 4 3 1, C = a 0 0 b. 5 3 0 0 b b 1 a b b 0 Example Problem : Gaussian elimination and matrix inversion [4] Points: (a)[1](m); (b)[1](m); (c)[1](m); (d)[1](m) (a) Solve the following system of linear equations using Gaussian elimination. [Check your result: the norm of x is x = 3.] 3x 1 + x x 3 = 1, x 1 x + 4x 3 =, x 1 + 1 x x 3 = 0. (b) How does the solution change when the last equation is removed? (c) What happens if the last equation is replaced by x 1 + 7 x x 3 = 0? (d) This system of equations can also be expressed in the form Ax = b. Calculate the inverse A of the 3 3 Matrix A using Gaussian elimination, and verify your answer to (a) using x = A b. Example Problem 3: Two-dimensional rotation matrices [4] Points: (a)[1](m); (b)[1](e); (c)[1](m); (d)[1](m) A rotation in two dimensions is a linear map, R : R R, that rotates every vector by a given angle about the origin without changing its length. (a) Find the dimensional rotation matrix R θ describing a rotation by the angle θ by proceeding R θ (e i ) as follows: Make a scetch that illustrates the effect e j e j of the rotation about the i axis on the three basis vectors e j (j = 1,, 3) (eg. for θ = π). The image vectors 6 e j of the basis vectors e j yield the columns of R θ. 1
(b) Write down the matrix R θi for the angles θ 1 = 0, θ = π/4, θ 3 = π/ and θ 4 = π. Compute the action of R θi (i = 1,, 3, 4) on a = (1, 0) T and b = (0, 1) T, and make a scetch to visualize the results. (c) The composition of two rotations again is a rotation. Show that R θ R φ = R θ+φ. Hint: Utilize the following addition theorems : cos(θ + φ) = cos θ cos φ sin θ sin φ, sin(θ + φ) = sin θ cos φ + cos θ sin φ. Remark: A geometric proof of these theorems (not requested here) follows from the figure by inspecting the three right-angled triangles with diagonals of length 1, cos φ and sin φ. cos θ cos φ θ φ sin θ cos φ cos φ 1 sin θ sin φ sin(θ + φ) π sin φ cos θ sin φ θ φ (d) Show that the rotation of an arbitrary vector r = (x, y) T by the angle θ does not change its length, i.e. that D θ r has the same length as r. θ cos(θ + φ) Example Problem 4: Basistransformation und lineare Abbildung in E [4] Points: (a)[0.5](m); (b)[0.5](m); (c)[0.5](e); (d)[0.5](e); (e)[0.5](m); (f)[0.5](m); (g)[0.5](e); (h)[0.5](e) Remark on notation: For this problem we denote vectors in euclidean space E using hats (e.g. ˆv j, ˆx, ŷ, ê 1 E ). Their components with respect to a given basis are vectors in R and are written without hats (e.g. x, y R ). Consider two bases for the Euclidean vector space E, one old {ˆv j }, and one new {ˆv i}, with ˆv 1 = 3 4 ˆv 1 + 1 3 ˆv, ˆv = 1 8 ˆv 1 + 1 ˆv. (a) The relation ˆv j = ˆv it i j expresses the old basis in terms of the new basis. Find the transformation matrix T = (t i j). (b) Find the matrix T, and use the inverse transformation ˆv j = ˆv i (t ) i j to express the new basis in terms of the old basis. (c) Let ˆx be a vector with components x = (1, ) T the new basis. [Kontrollergebnis: x = ( 1, 4 3 )T.] in the old basis. Find its components x in (d) Let ŷ by a vector with components y = ( 3 4, 1 3 )T in the new basis. Find its components y in the old basis.  (e) Let  be the linear map defined by ˆv 1 ˆv 1 and ˆv  ˆv. First find the matrix representation A of this map in the new basis, then use a basis transformation to find its matrix representation A in the old basis. [Check your result: (A) 1 = 3.] 5 (f) Let ẑ be the image vector onto which the vector ˆx is mapped by Â, i.e. ˆx  ẑ. Find its components z with respect to the new basis by using A, and its components z with respect to the old basis by using A. Are your results for z and z consistent? (g) Now make the choice ˆv 1 = ˆ3e 1 +ê and ˆv = 1 ê1+ 3 ê, for the old basis, where ê 1 = (1, 0) T and ê = (0, 1) T are the standard basis vectors on E. What are the components of ˆv 1, ˆv, ˆx and ẑ in the standard basis E?
(h) Make a sketch (with ê 1 and ê as unit vectors in the horizontal and vertical directions respectively), showing the old and new basis vectors, as well as the vectors ˆx and ẑ. Are the coordinates of these vectors, discussed in (c) and (f), consistent with your sketch? Example Problem 5: Basis transformations [6] Points: (a)[1](m); (b)[1](e); (c)[1](e); (d)[1](m); (e)[1](m); (f)[1](m) Consider the following three transformations in R 3, using the standard basis {e 1, e, e 3 }: A : Rotation about the third axis by the angle θ 3 = π, in the right-hand positive direction. 4 B : Dilation (stretching) of the first axis by the factor s 1 = 3; C : Rotation about the second axis by the angle θ = π, in the right-hand positive direction. Note: To understand what right-hand positive means, imagine wrapping your right hand around the axis of rotation, with your thumb pointing in the positive direction. Your other fingers will be curled in the direction of positive rotation. (a) Find the matrix representations (with respect to the standard basis) of A, B and C. (b) What is the image y = Bx of the vector x = (1, 1, 1) T under the dilation B? (c) What is the image z = Dx of x under the composition of all three maps, D = C B A? (d) Now consider a new basis {v j}, defined by a rotation of the standard basis by A, i.e. e A j v j. Draw the new and old basis vectors in the same figure. Find the transformation matrix T, and specify the matrix elements of the transformation between the old and the rotated bases using e j = v it i j. (e) In the {v i} basis let the vectors x and y be represented by x = v ix i and y = v iy i. Find the corresponding components x = (x 1, x, x 3 ) T and y = (y 1, y, y 3 ) T. (f) Let B denote the dilation B in the rotated basis. Find B by the appropriate transformation of the matrix B, and use the result to calculate the image y of x under B. (Does the result match that from (e)?) Example Problem 6: Triple Gaussian integral via transformation of variables [] Points: [](M) Calculate the following three-dimensional Gaussian integral (with a, b, c > 0, a, b, c R): ˆ I = dx dy dz e [a(x+y)+b(z y)+c(x z) ] R 3 Hint: Use the substitution u = a(x + y), v = b(z y), w = c(x z) and calculate the Jacobian determinant, using J = (u,v,w). You may use dx e x = π. [Check your result: if (x,y,z) a = b = c = π, then I = 1.] Example Problem 7: Matrix inversion [Bonus] Points: (a)[1](e); (b)[](m) Let M n be an n n matrix with matrix elements (M n ) i j = δ i jm + δ 1 j, with i, j = 1,..., n. 3
(a) Find the inverse matrices M and M 3. Verify in both cases that M n M n = 1. (b) Use the results from (a) to formulate a guess at the form of the inverse matrix Mn arbitrary n. Check your guess by calculating Mn M n. (c) Give a compact formula for the matrix elements (Mn ) i j. Check its validity by showing that (Mn ) i l (M n) l j = δ i j holds, by explicitly performing the sum on l. l for [Total Points for Example Problems: ] Homework Problem 1: Calculating determinants [4] Points: (a)[1](e); (b)[1,5](e); (c)[1,5](e) (a) Compute the derminant of the matrix D = ( 1 ) c 0 d 3. e (i) Which values must c and d have to ensure that det D = 0 for all values of e? (ii) Which values must d and e have to ensure that det D = 0 for all values of c? Could you have found the results of (i,ii) without explicitly calculating det D? Now consider the two matrices A = ( ) 1 3 1 und B = 6 6. 0 1 5 5 8 (b) Compute the product AB, as well as its determinant det(ab) and inverse (AB). (c) Compute the product BA, as well as its determinant det(ba) and inverse (BA). Is it possible to calculate the determinant and the inverse of A and B? Homework Problem : Gaussian elimination and matrix inversion [3] Points: (a)[1](e); (b)[1](e); (c)[1](m) Consider the linear system of equations Ax = b, with 8 3a 6a A = 6a 5 4 + 6a. (1) 4 + 6a 5 + 3a (a) For a = 1 3, use Gaussian elimination to compute the inverse matrix A. (Remark: It is advisable to avoid the occurrence of fractions until the left side has been brought into rowstep form.) Use the result to find the solution x for b = (4, 1, 1) T. [Check your result: the norm of x is x = 117/18.] (b) For which values of a can the matrix A not be inverted? (c) If A can be inverted, the system of equations Ax = b has a unique solution for every b, namely x = A b. If A can not be inverted, then either the solution is not unique, or no solution exists at all it depends on b which of these two cases arises. Decide this for b = (4, 1, 1) T and the values for a found in (b), and determine x, if possible. 4
Homework Problem 3: Three-dimensional rotation matrices [4] Points: (a)[1](e); (b)[1](e); (c)[1](e); (d)[0,5]; (e)[0,5](e); (f)[](bonus,a) Rotations in three dimensions are represented by 3 3 dimensional matrices. Let R θ (n) be the rotation matrix that describes a rotation by the angle θ about an axis whose direction is given by the unit vector n. (a) Find the three rotation matrices R θ (e i ) for rotations about the three coordinate axes e 1, e and e 3 explicitly, by proceeding as follows: Make a separate scetch for each of j = 1, and 3 R θ (e i ) that illustrates the effect e j e j of a rotation about the i axis on the three basis vectors e j (j = 1,, 3) (e.g. for θ = π). The image vectors 6 e j of the basis vectors e j yield the columns of the sought rotation matrix R θ. (b) It can be shown that for a general direction n = (n 1, n, n 3 ) T rotation axis have the following form: of matrix elements of the (R θ (n)) i j = δ ij cos θ + n i n j (1 cos θ) ɛ ijk n k sin θ (ɛ ijk = Levi-Civita-Tensor). Use this formula to find the three rotation matrices R θ (e i ) (i = 1,, 3) explictly. Are your results consistent with those from (a)? (c) Write down the following rotation matrices explicitly, and compute and scetch their effect on the vector v = (1, 0, 1) T : (i) A = R π (e 3 ), (ii) B = R π ( 1 (e 3 e 1 )). (d) Rotation matrices form a group. Use A and B from (c) to illustrate that this group is not commutative (in contrast to the two-dimensional case!). (e) Show that a general rotation matrix R satisfies the relation Tr(R) = 1 + cos θ, where the trace of a matrix R is defined by Tr(R) = i (R)i i. (f) The product of two rotation matrices is again a rotation matrix. Consider the product C = AB of the two matrices from (c), and find the corresponding unit vector n and rotation angle θ. Hint: these are uniquely defined only up to an arbitrary sign, since R θ (n) and R θ ( n) describe the same rotation. (To be concrete, fix this sign by choosing the component n positive.) θ and n i are fixed by the trace and the diagonal elements of the rotation matrix, respectively; their relativ sign is fixed by the off-diagonal elements. [Check your result: n = 1/ 3.] Homework Problem 4: Basis transformations in E [4] Points: (a)[0.5](m); (b)[0.5](m); (c)[0.5](e); (d)[0.5](e); (e)[0.5](m); (f)[0.5](m); (g)[0.5](e); (h)[0.5](e) Remark on notation: For this problem we denote vectors in euclidean space E using hats (e.g. ˆv j, ˆx, ŷ, ê 1 E ). Their components with respect to a given basis are vectors in R and are written without hats (e.g. x, y R ). Consider two bases for the Euclidean vector space E, one old {ˆv j }, and one new {ˆv i}, with ˆv 1 = 1 5 ˆv 1 + 3 5 ˆv, ˆv = 6 5 ˆv 1 + 5 ˆv. (a) The relation ˆv j = ˆv it i j expresses the old basis in terms of the new basis. Find the transformation matrix T = (t i j). 5
(b) Find the matrix T, and use the inverse transformation ˆv j = ˆv i (t ) i j to express the new basis in terms of the old basis. (c) Let ˆx be a vector with components x = (, 1 )T in the old basis. Find its components x in the new basis. (d) Let ŷ by a vector with components y = ( 3, 1) T in the new basis. Find its components y in the old basis. (e) Let  be the linear map defined by ˆv  1 1(ˆv 3 1 ˆv ) and ˆv  1(4ˆv 3 1 + ˆv ). First find the matrix representation A of this map in the old basis, then use a basis transformation to find its matrix representation A in the new basis. [Check your result: (A ) 1 =.] 3 (f) Let ẑ be the image vector onto which the vector ˆx is mapped by Â, i.e. ˆx  ẑ. Find its components z with respect to the old basis by using A, and its components z with respect to the new basis by using A. Are your results for z and z consistent? [Check your result: z = 1(5, 3 1)T.] (g) Now make the choice ˆv 1 = ê 1 + ê and ˆv = ê 1 ê for the old basis, where ê 1 = (1, 0) T and ê = (0, 1) T are the standard basis vectors on E. What are the components of ˆv 1, ˆv, ˆx and ẑ in the standard basis E? (h) Make a sketch (with ê 1 and ê as unit vectors in the horizontal and vertical directions respectively), showing the old and new basis vectors, as well as the vectors ˆx and ẑ. Are the coordinates of these vectors, discussed in (c) and (f), consistent with your sketch? Homework Problem 5: Basis transformations and linear maps [6] Points: (a)[1](m); (b)[1](e); (c)[1](e); (d)[1](m); (e)[1](m); (f)[1](m) Consider the following three linear transformations in R 3, using the standard basis {e 1, e, e 3 }. A : Rotation around the first axis by the angle θ 1 = π 3 rotation. in the right-handed sense, i.e. a left-handed B : Dilation of the first and second axes, by the factors s 1 = and s = 4 respectively. C : A reflection in the,3-plane. (a) Find the matrix representations (using the standard basis) of A, B, C. Which of these transformations commute with each other (i.e. for which pairs of matrices does T 1 T = T T 1 )? (b) What is the image y = CAx of the vector x = (1, 1, 1) T under the transformation CA? (c) Find the vector z, whose image under the composition of all three transformations, D = C B A, gives y. [Note: D = A B C.] (d) Now consider a new basis {v i}, defined by a rotation and reflection CA of the standard basis, v i CA e i. [Caution: in the sample the order was reversed!] Sketch the old and new bases in the same picture. [Note: The new basis vectors are a left handed system! Why?] Find the transformation matrix T, and specify the matrix elements of the transformation between the old and the new basis, with e j = v it i j. 6
(e) In the {v i}-basis let the vectors z and y be represented by z = v iz i and y = v iy i. Find the corresponding components z = (z 1, z, z 3 ) T and y = (y 1, y, y 3 ) T. (f) Let D denote the representation of D in the new Basis. Find D by an appropriate Transformation of the Matrix D, and use the result to find the image y of z under D. (Does the result match the one from (e)?). Homework Problem 6: Triple Lorentz integral via transformation of variables [] Points: [](M) Calculate the following triple Lorentz integral (with a, b, c, d > 0, a, b, c, d R): ˆ 1 I = dx dy dz R [(xd + y) 3 + a ] 1 [(y + z x) + b ] 1 [(y z) + c ] Hint: Use the change of variables u = (xd + y)/a, v = (y + z x)/b, w = (y z)/c and calculate the Jacobian determinant using J = (u,v,w) (x,y,z). You may use dx(x + 1) = π. [Check your result: if a = b = c = π, d =, then I = 1.] 5 Homework Problem 7: Matrixinversion (Bonus) [0] Points: (a)[1](e); (b)[](m) Let M n be an n n matrix with matrix elements (M n ) i j = m δ ij + δ i+1,j, with i, j = 1,..., n. (a) Find the inverse matrices M and M 3. Verify in both cases that M n M n = 1. (b) Use the results from (a) to formulate a guess at the form of the inverse matrix Mn arbitrary n. Check your guess by calculating Mn M n. (c) Give a compact formula for the matrix elements (Mn ) i j. Check its validity by showing that (Mn ) i l (M n) l j = δ i j holds, by explicitly performing the sum on l. l for [Total Points for Homework Problems: 3] 7