# Stochastik IV, Übungsblatt 4

# Aufgabe 2, (a)

# Spektralzerlegung von Matrix A

A <- cbind(c(5, -2), c(-2, 5))
spect.decomp <- eigen(A, symmetric = TRUE)
D <- diag(spect.decomp$values)
P <- spect.decomp$vectors

# Spuren

(trA <- sum(diag(A)))
(trD <- sum(diag(D)))
trA == trD

# Einige "Test-Berechnungen"

# (i) A = PDP'

P %*% D %*% t(P)  

# (ii) P orthogonale Matrix

sqrt(as.numeric(P[,1] %*% P[,1]))
sqrt(as.numeric(P[,2] %*% P[,2]))
as.numeric(P[,1] %*% P[,2])  # ~0 (rounding error)

# (ii) Ax = lambda x

A %*% P[,1] == as.matrix(D[1,1] * P[,1])
A %*% P[,2] == as.matrix(D[2,2] * P[,2])