Práctica de matrices

Creación de la matriz Y

Y<-matrix(data=c(30,20,36,24,40),
          ncol=1,
          byrow=TRUE)
colnames(Y)<-c("Y")
print(Y)

##       Y
## [1,] 30
## [2,] 20
## [3,] 36
## [4,] 24
## [5,] 40

Creación de la matriz X

X <- cbind(rep(1, 5), matrix(
  data = c(4, 10, 3, 8, 6, 11, 4, 9, 8, 12),
  nrow = 5,
  ncol = 2,
  byrow = FALSE
)) 
colnames(X)<-c("cte","X1","X2")
print(X)

##      cte X1 X2
## [1,]   1  4 11
## [2,]   1 10  4
## [3,]   1  3  9
## [4,]   1  8  8
## [5,]   1  6 12

Creación de la sigma matriz

XX<-t(X)%*%X 
print(XX)

##     cte  X1  X2
## cte   5  31  44
## X1   31 225 247
## X2   44 247 426

Creación de la matriz XY

XY<-t(X)%*%Y
print(XY)

##        Y
## cte  150
## X1   860
## X2  1406

Cálculo de beta

beta<-solve(XX,XY)
print(beta)

##              Y
## cte 20.3663921
## X1  -0.8191104
## X2   1.6718287

Cálculo de las matrices A, M, P

A<-solve(XX)%*%t(X) |> print()

##            [,1]        [,2]       [,3]        [,4]       [,5]
## cte  0.26886326  0.82009885  2.5172982 -0.59538715 -2.0108731
## X1  -0.04711697  0.03887974 -0.1960461  0.08105437  0.1232290
## X2   0.02537068 -0.09785832 -0.1252059  0.03327842  0.1644152

P<-X%*%solve(XX)%*%t(X) |> print()

##             [,1]        [,2]        [,3]        [,4]       [,5]
## [1,]  0.35947282 -0.10082372  0.35584843  0.09489292  0.2906096
## [2,] -0.10082372  0.81746293  0.05601318  0.34827018 -0.1209226
## [3,]  0.35584843  0.05601318  0.80230643 -0.05271829 -0.1614498
## [4,]  0.09489292  0.34827018 -0.05271829  0.31927512  0.2902801
## [5,]  0.29060956 -0.12092257 -0.16144975  0.29028007  0.7014827

M<-diag(5)-P |> print()

##             [,1]        [,2]        [,3]        [,4]       [,5]
## [1,]  0.35947282 -0.10082372  0.35584843  0.09489292  0.2906096
## [2,] -0.10082372  0.81746293  0.05601318  0.34827018 -0.1209226
## [3,]  0.35584843  0.05601318  0.80230643 -0.05271829 -0.1614498
## [4,]  0.09489292  0.34827018 -0.05271829  0.31927512  0.2902801
## [5,]  0.29060956 -0.12092257 -0.16144975  0.29028007  0.7014827

Propiedades de P y M

Idempotencia de P

iP<-(P%*%P-P) |> round(digits=2) |> print()

##      [,1] [,2] [,3] [,4] [,5]
## [1,]    0    0    0    0    0
## [2,]    0    0    0    0    0
## [3,]    0    0    0    0    0
## [4,]    0    0    0    0    0
## [5,]    0    0    0    0    0

Idempotencia de M

iM<-(M%*%M-M) |> round(digits=2) |> print()

##      [,1] [,2] [,3] [,4] [,5]
## [1,]    0    0    0    0    0
## [2,]    0    0    0    0    0
## [3,]    0    0    0    0    0
## [4,]    0    0    0    0    0
## [5,]    0    0    0    0    0

Simetría de P

sP<-(t(P)-P) |> round(digits=2) |> print()

##      [,1] [,2] [,3] [,4] [,5]
## [1,]    0    0    0    0    0
## [2,]    0    0    0    0    0
## [3,]    0    0    0    0    0
## [4,]    0    0    0    0    0
## [5,]    0    0    0    0    0

Simetría de M

sM<-(t(M)-M) |> round(digits=2) |> print()

##      [,1] [,2] [,3] [,4] [,5]
## [1,]    0    0    0    0    0
## [2,]    0    0    0    0    0
## [3,]    0    0    0    0    0
## [4,]    0    0    0    0    0
## [5,]    0    0    0    0    0

Ortogonales

MP=0

oP<-M%*%P |> round(digits=2) |> print()

##      [,1] [,2] [,3] [,4] [,5]
## [1,]    0    0    0    0    0
## [2,]    0    0    0    0    0
## [3,]    0    0    0    0    0
## [4,]    0    0    0    0    0
## [5,]    0    0    0    0    0

MX=0

oX<-M%*%X |> round(digits=2) |> print()

##      cte X1 X2
## [1,]   0  0  0
## [2,]   0  0  0
## [3,]   0  0  0
## [4,]   0  0  0
## [5,]   0  0  0

Proyección de X sobre X

pX<-(P%*%X-X) |> round(digits=2) |> print()

##      cte X1 X2
## [1,]   0  0  0
## [2,]   0  0  0
## [3,]   0  0  0
## [4,]   0  0  0
## [5,]   0  0  0