Eigen vectors in Different Subspaces

> file <- "C:/Cauldron/garage/R/soulcraft/Volatility/Learn/Wolfgang_Multivariate/Data/food.dat"
> x <- as.data.frame(read.table(file))
> y <- as.matrix(x[, -1])
> colnames(y) <- c("bread", "vegetables", "fruits", "meat", "poultry",
+     "milk", "wine")
> rownames(y) <- paste(c("MA", "EM", "CA"), c(rep("2", 3), rep("3",
+     3), rep("4", 3), rep("5", 3)), sep = "")
> print(y)
    bread vegetables fruits meat poultry milk wine
MA2   332        428    354 1437     526  247  427
EM2   293        559    388 1527     567  239  258
CA2   372        767    562 1948     927  235  433
MA3   406        563    341 1507     544  324  407
EM3   386        608    396 1501     558  319  363
CA3   438        843    689 2345    1148  243  341
MA4   534        660    367 1620     638  414  407
EM4   460        699    484 1856     762  400  416
CA4   385        789    621 2366    1149  304  282
MA5   655        776    423 1848     759  495  486
EM5   584        995    548 2056     893  518  319
CA5   515       1097    887 2630    1167  561  284
> y <- as.matrix(y)

> X1 <- t(y) %*% y
> X2 <- y %*% t(y)
> u1 <- eigen(X1)$vectors
> u2 <- eigen(X2)$vectors

> round(eigen(X1)$values)
[1] 68290407   422347   157951    39622    23110    11691     3630
> round(eigen(X2)$values)
 [1] 68290407   422347   157951    39622    23110    11691     3630        0
 [9]        0        0        0        0

> t2 <- y %*% u1
> for (i in 1:7) {
+     t2[, i] <- t2[, i]/sqrt((eigen(X1)$values[i]))
+ }
> t2[, 1:6]
          [,1]        [,2]         [,3]       [,4]        [,5]        [,6]
MA2 -0.2079395  0.15271493 -0.474731503 -0.3443228 -0.03508016 -0.25069072
EM2 -0.2202730 -0.03833216 -0.092547949 -0.3609908 -0.12751513  0.62140992
CA2 -0.2940033 -0.16502413 -0.264787773  0.2122301  0.70461162  0.04043664
MA3 -0.2230495  0.26402657 -0.209010156 -0.1819590 -0.03332130  0.19897205
EM3 -0.2248963  0.19011088 -0.058384161 -0.2373929  0.20213231  0.17329765
CA3 -0.3483473 -0.41212963 -0.180113734  0.3157197  0.11037028 -0.06505336
MA4 -0.2469810  0.36544361 -0.004663459  0.1610017 -0.17306485 -0.06621540
EM4 -0.2780885  0.14568799 -0.092310321 -0.1235053 -0.06710894 -0.19207224
CA4 -0.3454264 -0.44571719 -0.149146527  0.1966490 -0.60617001  0.07431698
MA5 -0.2859014  0.47049301  0.029787315  0.3689016 -0.12761532 -0.29321003
EM5 -0.3193368  0.17993222  0.523470683  0.2662707  0.08659110  0.41859643
CA5 -0.3984468 -0.25513862  0.557423475 -0.4796958  0.07874982 -0.40929272
> u2[, 1:6]
            [,1]        [,2]         [,3]       [,4]        [,5]        [,6]
 [1,] -0.2079395 -0.15271493  0.474731503 -0.3443228 -0.03508016 -0.25069072
 [2,] -0.2202730  0.03833216  0.092547949 -0.3609908 -0.12751513  0.62140992
 [3,] -0.2940033  0.16502413  0.264787773  0.2122301  0.70461162  0.04043664
 [4,] -0.2230495 -0.26402657  0.209010156 -0.1819590 -0.03332130  0.19897205
 [5,] -0.2248963 -0.19011088  0.058384161 -0.2373929  0.20213231  0.17329765
 [6,] -0.3483473  0.41212963  0.180113734  0.3157197  0.11037028 -0.06505336
 [7,] -0.2469810 -0.36544361  0.004663459  0.1610017 -0.17306485 -0.06621540
 [8,] -0.2780885 -0.14568799  0.092310321 -0.1235053 -0.06710894 -0.19207224
 [9,] -0.3454264  0.44571719  0.149146527  0.1966490 -0.60617001  0.07431698
[10,] -0.2859014 -0.47049301 -0.029787315  0.3689016 -0.12761532 -0.29321003
[11,] -0.3193368 -0.17993222 -0.523470683  0.2662707  0.08659110  0.41859643
[12,] -0.3984468  0.25513862 -0.557423475 -0.4796958  0.07874982 -0.40929272