#--------------------------# # Loesungen UeBlatt2 # #--------------------------# # Aufgabe 1 # # 1a) x = 2*(1:8) x A1 = matrix(x, ncol=4, nrow=2, byrow=TRUE) A1 y = rev(x) y A2 = matrix(y, ncol=2, nrow=4) A2 v1 = c(-2.2, 3.3, 1.1) v2 = c(4.56, -7.89, 1.23) v3 = c(800, 400, 200) A3 = rbind(v1,v2,v3) A3 # 1b) x = 100:1 x B1 = diag(x) B1 # Check: head(B1,10) tail(B1,10) B1[1:10,1:10] B1[90:100,90:100] # ok, passt alles y = rep(1:100, 100) y B2 = matrix(y, 100, 100) B2 # Check: head(B2,10) tail(B2,10) B2[1:10,1:10] B2[90:100,90:100] E = matrix(1,100,100) BU = upper.tri(E) BL = -lower.tri(E) B3 = BU+BL B3 # Check: head(B3) tail(B3) B3[1:10,1:10] B3[90:100,90:100] # passt alles # Aufgabe 2 # # 2a) n = 10 x = 1:n F = outer(x,x) F F = -2*pi*1i*F/n F = exp(F) F = F/sqrt(n) F # 2b) eigen(F)$values # nur 1,-1,i,-i # 2c) F2 = F %*% F F4 = F2 %*% F2 F4 zapsmall(F4) Re(zapsmall(F4)) # Einheitsmatrix Im(zapsmall(F4)) # Nullmatrix # 2d) barF = Conj(F) # komplex konjugierte Fadj = t(barF) # eigentlich nicht noetig, # da F_{jk} = F_{kj} Fadj %*% F zapsmall( Fadj%*%F ) zapsmall( F%*%Fadj ) # Aufgabe 3 v1 = c(1,1/2,1/3) v2 = c(1/2,1/3,1/4) v3 = c(1/3,1/4,1/5) A = rbind(v1,v2,v3) b = c(1,0,0) solve(A,b)