Chapter 2, Linear Algebra

# first make sure the `LinearAlgebra` package is loaded
using LinearAlgebra

# we have seen 1D arrays, aka column vectors
vc = [1, 2, 3]

# a (1 x 3) row vector. 
vr = [10 20 30] 

# inner product,
vr * vc

# outer product
vc * vr

vc = [10; 20; 30] # Create a column vector. 
               # Semi-colon is used to change row.

# Semi-colon is used to change rows.
# Space is used to seperate columns.
M = [1 2 3; 4 5 6; 7 8 9] 

# first index is row, 2nd is col
M[1,2] = 0 
M

# matrix times column vector
M * vc 

# row vector times matrix
vr * M 

# Transpose of a matrix using single qoute.
M' 

# Inverse of a matrix.
inv(M) 

# matrix times matrix
M * inv(M) 

# turn M into a symmetric matrix
M = (M + M') ./ 2 

# eigenvalue decomposition from the Linear Algebra package
w, U = eigen(M) 

# The columns of U are authonormal eigenvectors
U * U' 

# checking the Eigenvector decomposition theorem
U * Diagonal(w) * U' 

# positive definite matrix
A = [ 2 -1 0; -1 2 -1; 0 -1 2 ]  

eigen(A)

# square root for positive definite matrices
function matSqrt(M)
    w, U = eigen(M)
    return U * Diagonal(w).^(1/2) * U'
end

matSqrt(A)

matSqrt(A) * matSqrt(A)

using CUDAdrv; CUDAdrv.name(CuDevice(0))
using LinearAlgebra, Flux, CuArrays

function test_gpu(n=1000,m=10)
		M = rand(n,n) |> gpu
    R = zeros(n,n) |> gpu
    for k in 1:m
        # perform matrix multiplication
        R = M * M
    end
    return 1
end

@time test_gpu(8192,10)

function test_cpu(n=1000,m=10)
		M = rand(n,n) |> cpu
    R = zeros(n,n) |> cpu
    for k in 1:m
        # perform matrix multiplication
        R = M * M
    end
    return 1
end

@time test_cpu(8192,10)