CODEWITHSUNDEEP
pythonarraysnumpymatrixlinear-algebra
Addman
Addman

Reputation: 341

Finding upper/lower triangular form of arbitrary matrix n*n - python

every matrix can be written in upper or lower triangular form simply just by rotating the basis. Is there a simple routine in python (numpy) to do it? I was unable to find it and I cant believe that there is no such thing. To ilustrate it:

matrix = numpy.array([[a,b,c],
                      [d,e,f],
                      [g,h,i]])

to

matrix2 = numpy.array([[z,0,0],
                       [y,x,0],
                       [v,u,t]])

letters are floats. So how to make this change, but not simply just by zeroing numbers b, c and f, but by correct rotation of basis in the most simple way.

Thank you!

Upvotes: 3

Views: 1096

Answers (1)

Pascal Bugnion
Pascal Bugnion

Reputation: 4928

You are looking for Schur decomposition. Schur decomposition decomposes a matrix A as A = Q U Q^H, where U is an upper triangular matrix, Q is a unitary matrix (which effects the basis rotation) and Q^H is the Hermitian adjoint of Q.

import numpy as np
from scipy.linalg import schur

a = np.array([[ 1., 2., 3.], [4., 5., 6.], [7., 8., 9.]])
u, q = schur(a) # q is the unitary matrix, u is upper triangular

repr(u)
# array([[  1.61168440e+01,   4.89897949e+00,   1.58820582e-15],
#        [  0.00000000e+00,  -1.11684397e+00,  -1.11643184e-15],
#        [  0.00000000e+00,   0.00000000e+00,  -1.30367773e-15]])

Upvotes: 6

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