# Keeping track of coefficients in Gram-Schmidt¶

In [1]:
#keep
import numpy as np
import numpy.linalg as la

In [2]:
#keep
A = np.random.randn(3, 3)


Let's start from regular old (modified) Gram-Schmidt:

In [3]:
#keep

Q = np.zeros(A.shape)

q = A[:, 0]
Q[:, 0] = q/la.norm(q)

# -----------

q = A[:, 1]
coeff = np.dot(Q[:, 0], q)
q = q - coeff*Q[:, 0]
Q[:, 1] = q/la.norm(q)

# -----------

q = A[:, 2]
coeff = np.dot(Q[:, 0], q)
q = q - coeff*Q[:, 0]
coeff = np.dot(Q[:, 1], q)
q = q - coeff*Q[:, 1]
Q[:, 2] = q/la.norm(q)

In [4]:
#keep
Q.dot(Q.T)

Out[4]:
array([[  1.00000000e+00,  -2.77555756e-17,  -5.55111512e-17],
[ -2.77555756e-17,   1.00000000e+00,  -5.55111512e-17],
[ -5.55111512e-17,  -5.55111512e-17,   1.00000000e+00]])

Now we want to keep track of what vector got added to what other vector, in the style of an elimination matrix.

Let's call that matrix $R$.

• Would it be $A=QR$ or $A=RQ$? Why?
• Where are $R$'s nonzeros?
In [5]:
R = np.zeros((A.shape[0], A.shape[0]))

In [6]:
Q = np.zeros(A.shape)

q = A[:, 0]
Q[:, 0] = q/la.norm(q)

R[0,0] = la.norm(q)

# -----------

q = A[:, 1]
coeff = np.dot(Q[:, 0], q)
R[0,1] = coeff
q = q - coeff*Q[:, 0]
Q[:, 1] = q/la.norm(q)

R[1,1] = la.norm(q)

# -----------

q = A[:, 2]
coeff = np.dot(Q[:, 0], q)
R[0,2] = coeff
q = q - coeff*Q[:, 0]
coeff = np.dot(Q[:, 1], q)
R[1,2] = coeff
q = q- coeff*Q[:, 1]
Q[:, 2] = q/la.norm(q)

R[2,2] = la.norm(q)

In [7]:
R

Out[7]:
array([[ 0.81919017, -1.17431844, -1.02300235],
[ 0.        ,  1.11510507,  0.82496372],
[ 0.        ,  0.        ,  1.98359192]])
In [8]:
la.norm(Q.dot(R) - A)

Out[8]:
2.026584714835721e-16

This is called QR factorization.

• When does it break?
• Does it need something like pivoting?
• Can we use it for something?
In [ ]: