## ----echo=FALSE, include=FALSE------------------------------------------------
library(knitr)
opts_chunk$set(fig.width = 5, fig.height = 5, fig.cap = '', collapse = TRUE)


## ----app1---------------------------------------------------------------------
A <- matrix(1:4, nrow=2, byrow=TRUE)
B <- matrix(5:8, nrow=2, byrow=TRUE)
A + B


## ----app2---------------------------------------------------------------------
A <- matrix(c(1,-4,-2,5,3,-6), nrow=2)
B <- matrix(c(6,4,2,-5,-3,-1), nrow=3)
A %*% B
B %*% A


## ----app3---------------------------------------------------------------------
A <- matrix(1:6, nrow=2, byrow=TRUE)
A
t(A)


## ----app4---------------------------------------------------------------------
I <- matrix(0, ncol=2, nrow=2)
diag(I) <- 1
I

I <- matrix(0, ncol=5, nrow=5)
diag(I) <- 1
I


## ----app5---------------------------------------------------------------------
A <- matrix(1:4, nrow=2, byrow=TRUE)
Inv <- solve(A)
Inv

AA <- A %*% Inv
AA
round(AA, 10)


## ----app6---------------------------------------------------------------------
A <- matrix(1:4, nrow=2, byrow=TRUE)
B <- matrix(4:1, nrow=2, byrow=TRUE)
AB <- A %*% B
solve(AB)
# the same as
solve(B) %*% solve(A)


## ----app7---------------------------------------------------------------------
A <- matrix(c(3,2,4,-4), nrow=2)
b <- matrix(c(11, -6))
solve(A) %*% b


## ----app8---------------------------------------------------------------------
A <- matrix(c(.6,-.8,.8,.6), nrow=2)
s <- matrix(c(3, 4))
As <- A %*% s
round(As, 10)

S <- matrix(c(1,1,3,-2,0,5,-1,4,-2.5,-4), nrow=2)
AS <- A %*% S
AS


## ----app9---------------------------------------------------------------------
angle <- acos(A[1])
angle
# in degrees
180*angle/ pi


## ----app10--------------------------------------------------------------------
M <- matrix(c(3,2,4,-4), nrow=2)
eigen(M)

M <- matrix(c(1,3,3,2), nrow=2)
eigen(M)