## Loading required package: MASS## Loading required package: matrixcalcCreate a matrix \(A\) with \(p_1 = 10\) rows and \(p_2 = 9\) columns.
p1 <- 10 p2 <- 9 m1 <- 6 # number of rows in A11 m2 <- 3 # number of columns in A11 A <- matrix(rnorm(10, mean = 0, sd = 0.1), p1, p2)
Find the approximation low-rank of \(A\) under singular value decomposition (SVD)
j1 <- svd(A)$d j2 <- svd(A)$u j3 <- svd(A)$v num <-length(which(j1 != 0)) D <- diag(j1) D[num, num] <- 0 app_low_rank_A <- j2 A11 <- app_low_rank_A[1:m1, 1:m2] A12 <- app_low_rank_A[1:m1, (1+m2):p2] A21 <- app_low_rank_A[(1+m1):p1, 1:m2] A22 <- matrix(NA, nrow = p1 - m1, ncol = p2-m2) Arecovery <- rbind(cbind(A11, A12), cbind(A21, A22)) Arecovery
## [,1] [,2] [,3] [,4] [,5]
## [1,] -0.66011468 -0.74563894 0.02467862 -0.010085324 -0.026429084
## [2,] 0.09530799 -0.06679183 0.14749935 -0.982178325 0.001517307
## [3,] 0.46312641 -0.44958316 -0.49086217 0.002813528 0.584723258
## [4,] -0.16104826 0.14686398 -0.29543824 -0.070554121 -0.002563894
## [5,] -0.16095558 0.14690560 -0.29543961 -0.070553558 -0.002562418
## [6,] 0.24219851 -0.11041542 0.29423189 0.071047115 0.003855809
## [7,] 0.25905351 -0.21244434 -0.24583975 NA NA
## [8,] 0.24984942 -0.21657834 -0.24570292 NA NA
## [9,] -0.06617942 0.02507524 -0.07347428 NA NA
## [10,] 0.31026181 -0.29904350 0.59105240 NA NA
## [,6] [,7] [,8] [,9]
## [1,] -0.0097605613 0.0116298862 0.076485432 0.0065082700
## [2,] 0.0005603586 -0.0006676774 -0.004391066 -0.0003736430
## [3,] 0.0027229286 -0.0032444189 -0.021337335 -0.0018156286
## [4,] -0.9274494816 0.0011282190 0.007419876 0.0006313694
## [5,] 0.1518788419 0.9149391184 0.007415606 0.0006310060
## [6,] -0.1514011782 0.1788092701 0.884647755 -0.0009495087
## [7,] NA NA NA NA
## [8,] NA NA NA NA
## [9,] NA NA NA NA
## [10,] NA NA NA NAApply smc.FUN to recover the block \(A_{22}\)
A22 <- smc.FUN(Arecovery, 2, "True", m1, m2) A22
## [,1] [,2] [,3] [,4] [,5] [,6]
## [1,] -0.016783565 0.12708065 0.20959377 0.18223474 -0.29146769 2.846038e-16
## [2,] -0.016774224 0.12700992 0.20947712 0.18213331 -0.29130547 2.844454e-16
## [3,] -0.005016115 0.03798068 0.06264142 0.05446461 -0.08711113 8.505972e-17
## [4,] 0.040351353 -0.30552962 -0.50390917 -0.43813208 0.70075194 -6.842496e-16