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using Dates
include("jlFiles/printmat.jl")
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With a matrix $A$ and a scalar $c$, do
A*c (textbook: $Ac$) to multiply each element of $A$ by $c$
A .+ c (textbook: $A+cJ$, where $J$ is a matrix of ones) to add $c$ to each element of $A$
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A = [1 3;3 4]
c = 10
println("A:")
printmat(A)
println("c:")
printmat(c)
println("A*c:")
printmat(A*c)
println("A .+ c:")
printmat(A .+ c) #notice the dot in .+
With two matrices of the same dimensions ($A$ and $B$), do
A+B (textbook: $A+B$) to add them (element by element)
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A = [10;11] #A and B are 2x1 matrices
B = [2;5]
println("A:")
printmat(A)
println("B:")
printmat(B)
println("A-B:")
printmat(A-B)
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A = [1 3;3 4] #A and B are 2x2 matrices
B = [1 2;3 -2]
println("A:")
printmat(A)
println("B:")
printmat(B)
println("A+B:")
printmat(A+B)
Multiplying matrices ($A$ and $B$) of conformable dimensions
A*B (textbook: $AB$)
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A = [1 3;3 4] #A and B are 2x2 matrices
B = [1 2;3 -2]
println("A:")
printmat(A)
println("B:")
printmat(B)
println("A*B:")
printmat(A*B)
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A = [1 3;3 4] #A is 2x2, B is a vector with 2 elements
B = [2;5]
println("A:")
printmat(A)
println("B:")
printmat(B)
println("A*B:")
printmat(A*B)
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A = [10;11]
println("A: ")
printmat(A)
println("A': ")
printmat(A')
B = [1 2 3;4 5 6]
println("B: ")
printmat(B)
println("B': ")
printmat(B')
Inner and outer products of (column) vectors with $k$ elements:
x'z (textbook: $x'z$) to get a scalar and x*z' (textbook: $xz'$) to get a $k\times k$ matrix
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x = [10;11]
z = [2;5]
println("x and z")
printmat([x z])
println("x'z: ")
printlnPs(x'z)
println("x*z':")
printmat(x*z')
A quadratic form ($A$ is an $n \times n$ matrix and x is an $n$ vector)
x'A*x (textbook: $x'Ax$) to get a scalar
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A = [1 3;3 4]
println("x:")
printmat(x)
println("A:")
printmat(A)
println("x'A*x: ")
printlnPs(x'A*x)
A matrix inverse of an $nxn$ matrix $A$:
inv(A) or A^(-1) (textbook: $A^{-1}$)
The inverse is such that $AA^{-1}=I$ and $A^{-1}A=I$
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println("A:")
printmat(A)
println("inv(A):")
printmat(inv(A))
println("inv(A)*A:")
printmat(inv(A)*A)
$X'X$ or $\sum\nolimits_{t=1}^{T}x_{t}x_{t}^{\prime}$?
Let $x_t$ be a (column) vector with values of $K$ regressors for observation $t$. Then $x_{t}x_{t}^{\prime}$ is the outer product (a $K\times K$ matrix) and $\sum\nolimits_{t=1}^{T}x_{t}x_{t}^{\prime}$ is just the sum (of each element) across the $T$ observations.
We can calculate the same thing by (a) letting $X$ be a $T\times K$ matrix with $x_{t}^{\prime}$ in row $t$ and (b) then calculating $X'X$.
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x₁ = [1;-1] #(column vector)
x₂ = [1;0]
x₃ = [1;1.0]
X = [x₁';x₂';x₃']
println("X")
printmat(X)
(T,K) = (size(X,1),size(X,2))
Sxx1 = x₁*x₁' + x₂*x₂' + x₃*x₃' #just to illustrate
Sxx2 = zeros(K,K)
for t = 1:T
Sxx2 = Sxx2 + X[t,:]*X[t,:]' #yes, X[t,:] becomes a column vector
end
Sxx3 = X'X
println("sum of outer products, three versions")
printmat(Sxx1)
printmat(Sxx2)
printmat(Sxx3)
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