Ejercicio 1: Elipses


a) Sean \(D\) una matriz diagonal en \(R^{2x2}\), \(D=\left(\begin{array}{cc} 2 & 0\\0 & 3\end{array}\right)\) , \(x\in R^2\) y \(c=1\). Graficamos la elipse \(x^tDx=c^2\). 


tita<-seq(0,2*pi,2*pi/100)
x1<-(1/sqrt(2))*cos(tita)
x2<-(1/sqrt(3))*sin(tita)

par(pty="s")
plot(x1,x2,type="l",main=expression(paste(x^t,D,x,"=",c^2,sep="")))
points(0,0,pch=15)


b) Sean \(A\) una matriz simétrica y definida positiva en \(R^{2x2}\), \(A=\left(\begin{array}{cc} 2.5 & -0.5\\-0.5 & 2.5\end{array}\right)\) , \(x\in R^2\) y \(c=2\). Graficamos la elipse \(x^tAx=c^2\). 


A<-matrix(c(2.5,-0.5,-0.5,2.5),nrow=2)
eig<-eigen(A)
autovalores<-eig$values
U<-eig$vectors
x1<-vector()
x2<-vector()

for (i in 1:length(tita)){
  y1<-(2/sqrt(3))*cos(tita[i])
  y2<-(2/sqrt(2))*sin(tita[i])
  V<-U%*%c(y1,y2)
  x1[i]<-V[1]
  x2[i]<-V[2]
}

par(pty="s")
plot(x1,x2,type="l",col="red",main=expression(paste(x^t,A,x,"=",c^2,sep="")))
points(0,0,pch=15)


c) Sean \(A\) una matriz simétrica y definida positiva en \(R^{2x2}\), \(A=\left(\begin{array}{cc} 2.5 & -0.5\\-0.5 & 2.5\end{array}\right)\) , \(x\in R^2\) , \(c=2\) y \(b=(1,3)^t\). Graficamos la elipse \((x-b)^tA(x-b)=c^2\). 


b<-c(1,3)
x1<-x1+rep(b[1],length(x1))# x1+1
x2<-x2+rep(b[2],length(x2))# x2+3

par(pty="s")
plot(x1,x2,type="l",col="green",main=expression(paste((x-b)^t,A,(x-b),"=",c^2,sep="")))
points(1,3,pch=15)