Practica Matrices

setup, include=FALSE} knitr::opts_chunk$set(echo = TRUE)

Creación de la matriz Y

Y <- matrix(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

Creacion 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

Creacion 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

Creacion de la matriz XY

# Creacion de la Matriz XY
XY <- t(X)%*%Y
print(XY)

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

Calculando BETA

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

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

Calculando A, P y M

A <- solve(XX)%*%t(X)
P <- X%*%solve(XX)%*%t(X)
M <- diag(5)-P

Demostracion numerica de matriz P es idempotente

round(P%*%P-P,2)

##      [,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

Demostracion numerica de matriz P es simetrica

round(t(P)-P, 2)

##      [,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

Demostracion numerica de P ortogonal

round(M%*%P,2)

##      [,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

Demostracion numerica de proyección de X sobre X

round(P%*%X-X, 2)

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

```