Linear Algebra Exercises in R

Section 1-2

Let B and A be a 2 by 2 matrix and ‘a’ a numeric constant as following.
Note: function runif(n, min, max), generates n numbers of random values using uniform distribution in max and min boundary.

A <- matrix(runif(2*2, min=0, max=10), ncol=2)
B <- matrix(runif(2*2, min=0, max=10), ncol=2)
a <- runif(1)

print(A)

##      [,1] [,2]
## [1,]  6.0  8.0
## [2,]  8.7  3.8

print(B)

##      [,1] [,2]
## [1,]  1.7  2.5
## [2,]  1.3  6.4

print(a)

## [1] 0.85

Theorems

• 1. Check if \(det(A) = det(t(A))\)

all.equal(det(A), det(t(A)))

## [1] TRUE

• 2. Check if \(det(a*A) = a^{n}*det(A)\)

all.equal(det(a*A), (a^nrow(A))*det(A))

## [1] TRUE

• 3. Check if \(det(A*B) = det(A)*det(B)\)

Note: %*% represents matrix multiplication.

all.equal(det(A%*%B), det(A)*det(B))

## [1] TRUE

• 4. Check if \(det(A) = Conj(det(H(A)))\)

For evaluating last theorem, we need to construct a complex matrix.
Lets sum A and a complex matrix which only contains imaginary values.

A <- A + matrix(1i*runif(2*2, min=0, max=10), ncol=2)
print(A)

##          [,1]     [,2]
## [1,] 6.0+0.1i 8.0+4.3i
## [2,] 8.7+0.3i 3.8+9.9i

Since R does not have H(x) function, knowing that \(H(A) = Conj(t(A))\), we can construct:

H <- function(x){
  Conj(t(x))
}

# r-base::det function does not defined for complex matrices
# thus we need to use Det function from 'complexplus' package
suppressMessages(require(complexplus))
all.equal(Det(A), Conj(Det(H(A))))

## [1] TRUE