Eigenvector mystery
10-18-2020, 11:58 PM (This post was last modified: 10-19-2020 01:32 AM by Albert Chan.)
Post: #3
 Albert Chan Senior Member Posts: 1,989 Joined: Jul 2018
RE: Eigenvector mystery
All eigenvectors are correct. The difference is only a scaling factor.

XCas> m := [[-4,-17], [2,2]]
XCas> eigenvalues(m)﻿ ﻿ ﻿ ﻿ ﻿ ﻿ → (-1-5*i,-1+5*i)

Solving both vectors at the same time, λ = -1±5i

m * [x,y] = λ * [x,y]
(m - λ) * [x,y] = 0

m - λ = $$\left(\begin{array}{cc} -3∓5i & -17 \\ 2 & 3∓5i \end{array}\right)$$

If you pick 2nd row, you get results of Wolfram Alpha

2 x + (3∓5i) y = 0
2 x = (-3±5i) y ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ → [x, y] = [-3±5i, 2] * t, where t is the parameter

If you pick 1st row, you get [x,y] = [-17, 3±5i] * t'
Confirming numpy's result (which is probably what Julia use)

>>> import numpy
>>> k, v = numpy.linalg.eig([[-4,-17],[2,2]])
>>> print k
[-1.+5.j -1.-5.j]
>>> print v
[[ 0.94590530+0.j ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ 0.94590530+0.j﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ]
﻿ [-0.16692447-0.27820744j﻿ ﻿ -0.16692447+0.27820744j]]
>>> print v * (-17/v[0][0])
[[-17.+0.j﻿ ﻿ -17.+0.j]
﻿ [ ﻿ 3.+5.j﻿ ﻿ ﻿ ﻿ 3.-5.j]]

Update: numpy returns normalized eigenvectors (length = 1)
>>> sum(abs(v)**2) ** 0.5 ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ # numpy eigenvectors are column vectors
array([ 1., 1.])
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 Messages In This Thread Eigenvector mystery - John Keith - 10-18-2020, 09:31 PM RE: Eigenvector mystery - pinkman - 10-18-2020, 10:54 PM RE: Eigenvector mystery - JurgenRo - 10-19-2020, 06:33 PM RE: Eigenvector mystery - Albert Chan - 10-19-2020, 07:13 PM RE: Eigenvector mystery - Thomas Okken - 10-19-2020, 07:19 PM RE: Eigenvector mystery - JurgenRo - 10-19-2020, 07:35 PM RE: Eigenvector mystery - JurgenRo - 10-19-2020, 07:26 PM RE: Eigenvector mystery - pinkman - 10-19-2020, 08:57 PM RE: Eigenvector mystery - Thomas Okken - 10-19-2020, 09:42 PM RE: Eigenvector mystery - Valentin Albillo - 10-20-2020, 12:08 AM RE: Eigenvector mystery - JurgenRo - 10-21-2020, 06:57 PM RE: Eigenvector mystery - JurgenRo - 10-21-2020, 06:58 PM RE: Eigenvector mystery - Albert Chan - 10-18-2020 11:58 PM RE: Eigenvector mystery - Michael de Estrada - 10-20-2020, 11:16 PM RE: Eigenvector mystery - John Keith - 10-24-2020, 02:03 PM RE: Eigenvector mystery - Albert Chan - 10-26-2020, 04:25 PM RE: Eigenvector mystery - John Keith - 10-27-2020, 12:37 PM

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