2023-02-13 Linear Algebra
Contents
2023-02-13 Linear Algebra#
Last time#
Forward and backward stability
Beyond IEEE double precision
Today#
Algebra of linear transformations
Polynomial evaluation and fitting
Orthogonality
using Plots
default(linewidth=4, legendfontsize=12)
Matrices as linear transformations#
Linear algebra is the study of linear transformations on vectors, which represent points in a finite dimensional space. The matrix-vector product \(y = A x\) is a linear combination of the columns of \(A\). The familiar definition,
can also be viewed as
Math and Julia Notation#
The notation \(A_{i,j}\) corresponds to the Julia syntax A[i,j] and the colon : means the entire range (row or column). So \(A_{:,j}\) is the \(j\)th column and \(A_{i,:}\) is the \(i\)th row. The corresponding Julia syntax is A[:,j] and A[i,:].
Julia has syntax for row vectors, column vectors, and arrays.
[1. 2 3; 4 5 6]
# np.array([[1, 2, 3], [4, 5, 6]])
2×3 Matrix{Float64}:
1.0 2.0 3.0
4.0 5.0 6.0
[1 2 ; 4 3]
2×2 Matrix{Int64}:
1 2
4 3
[1 0; 0 2; 10 3]
3×2 Matrix{Int64}:
1 0
0 2
10 3
[1; 2 + 1im; 3]' # transpose
1×3 adjoint(::Vector{Complex{Int64}}) with eltype Complex{Int64}:
1+0im 2-1im 3+0im
Implementing multiplication by row#
function matmult1(A, x)
m, n = size(A)
y = zeros(m)
for i in 1:m
for j in 1:n
y[i] += A[i,j] * x[j]
end
end
y
end
A = reshape(1.:12, 3, 4) # 3x4 matrix
#x = [10., 0, 0, 0]
#matmult1(A, x)
3×4 reshape(::StepRangeLen{Float64, Base.TwicePrecision{Float64}, Base.TwicePrecision{Float64}, Int64}, 3, 4) with eltype Float64:
1.0 4.0 7.0 10.0
2.0 5.0 8.0 11.0
3.0 6.0 9.0 12.0
# Dot product
A[2, :]' * x
UndefVarError: x not defined
Stacktrace:
[1] top-level scope
@ In[7]:2
[2] eval
@ ./boot.jl:368 [inlined]
[3] include_string(mapexpr::typeof(REPL.softscope), mod::Module, code::String, filename::String)
@ Base ./loading.jl:1428
function matmult2(A, x)
m, n = size(A)
y = zeros(m)
for i in 1:m
y[i] = A[i,:]' * x
end
y
end
matmult2(A, x)
3-element Vector{Float64}:
10.0
20.0
30.0
Implementing multiplication by column#
function matmult3(A, x)
m, n = size(A)
y = zeros(m)
for j in 1:n
y += A[:, j] * x[j]
end
y
end
matmult3(A, x)
3-element Vector{Float64}:
10.0
20.0
30.0
A * x # We'll use this version
3-element Vector{Float64}:
10.0
20.0
30.0
Polynomial evaluation is (continuous) linear algebra#
We can evaluate polynomials using matrix-vector multiplication. For example,
using Polynomials
P(x) = Polynomial(x)
p = [0, -3, 0, 5]
q = [1, 2, 3, 4]
f = P(p) + P(q)
@show f
@show P(p+q)
x = [0., 1, 2]
f.(x)
f = Polynomial(1 - x + 3*x^2 + 9*x^3)
P(p + q) = Polynomial(1 - x + 3*x^2 + 9*x^3)
3-element Vector{Float64}:
1.0
12.0
83.0
plot(f, legend=:bottomright, xlim=(-2, 2))
Polynomial evaluation is (discrete) linear algebra#
V = [one.(x) x x.^2 x.^3]
3×4 Matrix{Float64}:
1.0 0.0 0.0 0.0
1.0 1.0 1.0 1.0
1.0 2.0 4.0 8.0
V * p + V * q
3-element Vector{Float64}:
1.0
12.0
83.0
V * (p + q)
3-element Vector{Float64}:
1.0
12.0
83.0
Vandermonde matrices#
A Vandermonde matrix is one whose columns are functions evaluated at discrete points.
function vander(x, k=nothing)
if isnothing(k)
k = length(x)
end
m = length(x)
V = ones(m, k)
for j in 2:k
V[:, j] = V[:, j-1] .* x
end
V
end
vander (generic function with 2 methods)
x = LinRange(-1, 1, 50)
V = vander(x, 4)
scatter(x, V, legend=:bottomright)
Fitting is linear algebra#
x1 = [-.9, 0.1, .5, .8]
y1 = [1, 2.4, -.2, 1.3]
scatter(x1, y1, markersize=8)
V = vander(x1)
@show size(V)
p = V \ y1 # write y1 in the polynomial basis
scatter(x1, y1, markersize=8, xlims=(-1, 1))
#plot!(P(p), label="P(p)")
plot!(x, vander(x, 4) * p, label="\$ V(x) p\$", linestyle=:dash)
size(V) = (4, 4)
Some common terminology#
The range of \(A\) is the space spanned by its columns. This definition coincides with the range of a function \(f(x)\) when \(f(x) = A x\).
The (right) nullspace of \(A\) is the space of vectors \(x\) such that \(A x = 0\).
The rank of \(A\) is the dimension of its range.
A matrix has full rank if the nullspace of either \(A\) or \(A^T\) is empty (only the 0 vector). Equivalently, if all the columns of \(A\) (or \(A^T\)) are linearly independent.
A nonsingular (or invertible) matrix is a square matrix of full rank. We call the inverse \(A^{-1}\) and it satisfies \(A^{-1} A = A A^{-1} = I\).
\(\DeclareMathOperator{\rank}{rank} \DeclareMathOperator{\null}{null} \) If \(A \in \mathbb{R}^{m\times m}\), which of these doesn’t belong?
\(A\) has an inverse \(A^{-1}\)
\(\rank (A) = m\)
\(\null(A) = \{0\}\)
\(A A^T = A^T A\)
\(\det(A) \ne 0\)
\(A x = 0\) implies that \(x = 0\)
A = rand(4, 4)
B = A' * A - A * A'
@show B
det(A)
B = [0.2859982396934353 0.18759449524491867 -0.7832430558205019 1.2373350539578114; 0.18759449524491867 0.21064667505832912 -0.2917719197468648 0.966000242620058; -0.7832430558205019 -0.2917719197468648 -1.9014318752756914 -0.15443864588241019; 1.2373350539578114 0.966000242620058 -0.15443864588241019 1.404786960523927]
-0.03744462855916495
What is an inverse?#
When we write \(x = A^{-1} y\), we mean that \(x\) is the unique vector such that \(A x = y\). (It is rare that we explicitly compute a matrix \(A^{-1}\), though it’s not as “bad” as people may have told you.) A vector \(y\) is equivalent to \(\sum_i e_i y_i\) where \(e_i\) are columns of the identity. Meanwhile, \(x = A^{-1} y\) means that we are expressing that same vector \(y\) in the basis of the columns of \(A\), i.e., \(\sum_i A_{:,i} x_i\).
using LinearAlgebra
A = rand(4, 4)
4×4 Matrix{Float64}:
0.979985 0.360392 0.184604 0.846419
0.897927 0.207376 0.578949 0.852997
0.684921 0.88859 0.672726 0.080342
0.513169 0.0925713 0.0562695 0.587593
A \ A
4×4 Matrix{Float64}:
1.0 0.0 0.0 2.93245e-16
3.06357e-16 1.0 0.0 -1.91637e-16
5.05927e-17 0.0 1.0 2.02222e-17
2.18191e-16 0.0 0.0 1.0
inv(A) * A
4×4 Matrix{Float64}:
1.0 -2.48484e-16 -3.63082e-16 1.05015e-15
1.1956e-16 1.0 -2.93023e-16 -1.03256e-15
-3.04166e-17 1.22352e-16 1.0 2.82565e-16
2.55657e-16 -1.42748e-16 -1.54536e-16 1.0
Inner products and orthogonality#
The inner product
Examples with inner products#
x = [0, 1]
y = [1, 1]
@show x' * y
@show y' * x;
x' * y = 1
y' * x = 1
ϕ = pi/6
y = [cos(ϕ), sin(ϕ)]
cos_θ = x'*y / (norm(x) * norm(y))
@show cos_θ
@show cos(ϕ-pi/2);
cos_θ = 0.49999999999999994
cos(ϕ - pi / 2) = 0.4999999999999999
Polynomials can be orthogonal too!#
x = LinRange(-1, 1, 50)
A = vander(x, 4)
M = A * [.5 0 0 0; # 0.5
0 1 0 0; # x
0 0 1 0]' # x^2
scatter(x, M)
plot!(x, 0*x, label=:none, color=:black)
Which inner product will be zero?
Which functions are even and odd?
Polynomial inner products#
M[:,1]' * M[:,2]
-2.220446049250313e-16
M[:,1]' * M[:,3]
8.673469387755102
M[:,2]' * M[:,3]
-4.440892098500626e-16