GeoLing12: The product of matrices. C ¯ ontents: • Multiplying
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GeoLing12: The product of matrices. C ¯ ontents: • Multiplying
Geometria Lingotto. GeoLing12: The product of matrices. Contents: ¯ • Multiplying matrices. Recommended lecture: Leling 9. ¯ EXERCISES 1. Compute the following products: (a) 1 2 0 1 1 · (c) 0 −1 −1 1 1 · 2 3 0 −3 5 1 3 4 −2 1 ; (b) 4 3 3 1 1 · 0 −1 −1 1 1 ; (d) 4 3 3 1 1 · 1 2 0 1 1 2 3 4 −2 1 (a) N= 0 1 0 0 (b) 0 0 N = 0 0 1 0 0 0 0 0 1 0 0 0 0 0 3. Calculate N 3 = N · N · N : (a) 0 1 0 N = 0 0 1 0 0 0 Ingegneria dell’Autoveicolo, GL12 1 (b) 0 0 N = 0 0 1 0 0 0 0 1 0 0 2 2 1 3 3 3 0 4 4 −3 −2 −2 5 1 1 2. Calculate N 2 = N · N : 0 0 1 0 Geometria Geometria Lingotto. 4. Is there a matrix X such that A · X = B ? (a) A= 1 2 0 1 1 (b) A = 1 1 1 (c) A = 1 2 2 2 0 2 2 1 3 1 1 , B= 4 −2 1 0 1 2 0 , B= 1 2 0 1 0 0 1 0 1 , B= −1 1 5. Compute A · At : (a) A= 0 1 −1 0 (b) A= 1 3 2 −1 (c) A= cos(θ) − sin(θ) sin(θ) cos(θ) 6. Prove that A · At is a symmetric matrix. 7. Take A ∈ R2,2 and prove that if A · At = 0 than A = 0. Is the same true for A ∈ Z2,2 2 ?. 8. Is it true that ρ(A · B) = min{ρ(A), ρ(B)} ?. 9. Matrix multiplication according to Strassen. Check that a11 a12 a21 a22 b11 b12 c11 c12 · = b21 b22 c21 c22 c11 c12 , dove c21 c22 = p1 + p4 − p5 + p 7 , = p3 + p5 , = p2 + p4 , = p1 + p3 − p2 + p 6 . p1 = (a11 + a22 )(b11 + b22 ), p2 = (a21 + a22 )b11 , p3 = a11 (b12 − b22 ), 1 2 p4 = a22 (b21 − b11 ), and . Multiply according to Strassen · 3 4 p5 = (a11 + a12 )b22 , p = (a21 − a11 )(b11 + b12 ), 6 p7 = (a12 − a22 )(b21 + b22 ). Ingegneria dell’Autoveicolo, GL12 2 Geometria Geometria Lingotto. 5 6 7 8 10. Complex numbers as matrices. Let J = 0 −1 1 0 . (a) Check that J is a square root of −1, i.e. J 2 = −1, with −1 = −I = −I2 . 0 x −y x −y 0 0 0 0 (b) Let z = xI + yJ = and z = x I + y J = . Calculate y x y 0 x0 z · z0 . (c) Verify z t = xI − yJ and z · z t = (x2 + y 2 )I . 0 1 0 0 0 0 1 0 0 −1 0 0 0 0 0 0 1 0 11. Quaternions. Let i = 0 0 0 −1 , j = −1 0 0 0 , k = 0 0 0 1 0 0 −1 0 0 −1 Verify the following relations, discovered by Hamilton, hold: 0 0 1 0 −1 0 . 1 0 0 0 0 0 i2 = j 2 = k 2 = ijk = −1 . These formulas are engraved on one of Dublin’s bridges, cf. http://en.wikipedia.org/wiki/Quaternion A quaternion q is a matrix: a b c d −b a −d c = q= −c d a −b −d −c b a 1 0 = a 0 0 0 1 0 0 0 0 1 0 0 0 0 −1 +b 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 +c 0 −1 −1 0 0 −1 1 0 1 0 0 0 0 0 1 0 +d 0 0 0 −1 Check that a = 0 and b2 + c2 + d2 = 1 imply q 2 = −1. Ingegneria dell’Autoveicolo, GL12 3 Geometria 0 0 1 0 −1 0 . 1 0 0 0 0 0