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## R script for STAT 893 - LA Review II
## First problem set
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options(digits=4)
## create matrix A

A<-matrix(c(1,1,-5,1,1,2,3,1,4,0,-1,2,-1,3,-1),nrow=3,ncol=5)

square<-A%*%t(A);square
eigen(square)  			## prints both eigenvalues and eigenvectors (as columns)
P<-eigen(square)$vectors 	## NOTE: R automatically chooses normalized eigenvectors

new<-t(P)%*%square%*%P		## new = P'(AA')P

## cleaning up round-off errors (not necessary)
for (i in 1:nrow(P))
for (j in 1:ncol(P))
if (abs(new[i,j])<1e-12) new[i,j]<-0
new  					## displays P'(AA')P


## Now, the reverse direction

square<-t(A)%*%A;square
eigen(square)  			## prints both eigenvalues and eigenvectors (as columns)
P<-eigen(square)$vectors  	## NOTE: R automatically chooses normalized eigenvectors

new<-t(P)%*%square%*%P		## now, new = P'(A'A )P (different P, obviously)

for (i in 1:nrow(P))
for (j in 1:ncol(P))
if (abs(new[i,j])<1e-12) new[i,j]<-0
new