mat <- matrix(c(1,1,1,1,3,2,1,2,2), 3, 3)
eigens <- eigen(mat)
evs <- eigens$values
evecs <- eigens$vectors
a <- evs[1]
b <- evs[2]
c <- evs[3]
x0 <- 5
y0 <- 10
z0 <- 3
alpha <- atan(evecs[ , 1][2] / evecs[ , 1][1])
theta <- seq(0, 2 * pi, length=(1000))
# See Timm, Multivariate Analysis, page 62, Figure 1.6.2
# See Lay, Linear Algebra, third edition, page 84
x <- x0 + a * cos(theta) * cos(alpha) - b * sin(theta) * sin(alpha)
y <- y0 + a * cos(theta) * sin(alpha) + b * sin(theta) * cos(alpha)
z <- z0 + a * cos(theta) * sin(alpha) + c * sin(theta) * cos(alpha)
plot(x, y, z, type = "l", main = expression("x y z"), asp = 1, xlim=c(-5,20),ylim=c(-5,20), zlim=-5,20,xaxs = 'i', yaxs = 'i', zaxs='i')

    
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