(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 $b$:
import numpy as np
import matplotlib.pyplot as plt
import matplotlib.cm as cm
%matplotlib inline
A = np.array([[1, 2],[0, 1]])
n=21
colors = cm.viridis(np.linspace(0, 1, n))
θ = np.linspace(0,2*np.pi,n)
x_0 = np.cos(θ)
x_1 = np.sin(θ)
ϕ_0 = np.zeros(len(x_0))
ϕ_1 = np.zeros(len(x_1))
for i in range(len(x_0)):
x = np.array([x_0[i], x_1[i]])
ϕ = A.dot(x)
ϕ_0[i] = ϕ[0]
ϕ_1[i] = ϕ[1]
#------------------- plot result
plt.figure(figsize=(5,5))
plt.rc('font', size=14)
plt.plot(x_0,x_1, '-', color='gray', label='x') # x
plt.scatter(x_0,x_1, color=colors, s=50, label='') # x
plt.plot(ϕ_0,ϕ_1, ':', color='gray', label=r'Ax') # x
plt.scatter(ϕ_0,ϕ_1, color=colors, s=50, label='') # Ax
plt.xticks([])
plt.yticks([])
plt.legend(frameon=False)
plt.xlim([-2.5,2.5])
plt.ylim([-2.5,2.5]);
$Av = \lambda v$
$AV = V\Lambda$.
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):
import numpy as np
A = np.array([[1,1],[1,1.0001]])
b = np.array([2,2.0000])
x = np.linalg.solve(A,b)
print('x = ', x)
print('Condition number of A = ', np.linalg.cond(A))
x = [2. 0.] Condition number of A = 40002.00007491187
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): $$RE_x = C(A)\cdot RE_b.$$