(2) x acts on A to produce b
For matrix-matrix multiplication, this process is repeated:
* Here, the columns of $C$ are linear combinations of the columns of $A$, where elements of $B$ are the multipliers in those linear combinations.Also:
* Here, the rows of $C$ are linear combinations of the rows of $A$, where elements of $B$ are the multipliers in those linear combinations.Identities:
Consider 2D plane:
Numerically singular matrices are almost singular. That is, they may be singular to within roundoff error, or the near singularity may result in inaccurate results.
$Ax=b$
$x=A^{-1}b$.
$A^{-1}b \rightarrow$ change basis of $b$:
using Plots
using ColorSchemes
A = [1 2; 0 1]
n=21
colors=[get(ColorSchemes.viridis, i) for i in LinRange(0,1,n)]
θ = LinRange(0,2π,n)
x_0 = cos.(θ)
x_1 = sin.(θ)
ϕ_0 = zeros(length(x_0))
ϕ_1 = zeros(length(x_1))
for i in 1:length(x_0)
x = [x_0[i], x_1[i]]
ϕ = A*x
ϕ_0[i] = ϕ[1]
ϕ_1[i] = ϕ[2]
end
#------------------- plot result
Plots.resetfontsizes(); Plots.scalefontsizes(1.5)
plot( x_0,x_1, color="gray", label="x") # x
scatter!(x_0,x_1, label=nothing, color=colors, ms=6) # x
plot!( ϕ_0,ϕ_1, linestyle=:dash, color="gray", label="Ax") # x
scatter!(ϕ_0,ϕ_1, label=nothing, color=colors, ms=6) # Ax
plot!(xticks=nothing, yticks=nothing, foreground_color_legend=nothing)
plot!(xlim=[-2.5,2.5], ylim=[-2.5,2.5])
plot!(aspect_ratio=:equal)
$Av = \lambda v$
$AV = V\Lambda$.
So, $Ax=b$ can be written as $\Lambda\hat{x} = \hat{b}$.
We want to know the sizes of vectors and matrices to get a feel for scale.
This is important in many contexts, including stability analysis and error estimation.
A norm provides a positive scalar as a measure of the length.
Notation is $\|x\|$ for the norm of $x$.
Properties:
P-norms:
Consider the "unit balls" of these norms for vectors in the plane.
Matrix norm (induced):
using LinearAlgebra
A = [1 1; 1 1.0001]
b = [2,2.0000]
x = A\b
println("x = $x")
println("Condition number of A = $(cond(A))")
x = [2.0, 0.0] Condition number of A = 40002.000074915224
Note, a property of norms gives $\|Ax\| \le \|A\|\|x\|$
So,
$$\frac{\|\delta x\|}{\|x\|} \le \|A\|\|A^{-1}\|\frac{\|\delta b\|}{\|b\|}$$In terms of relative error (RE): $$RE_x = C(A)\cdot RE_b.$$