Eigenvector mystery
10-21-2020, 06:57 PM
Post: #13
 JurgenRo Member Posts: 176 Joined: Jul 2015
RE: Eigenvector mystery
(10-20-2020 12:08 AM)Valentin Albillo Wrote:
(10-19-2020 09:42 PM)Thomas Okken Wrote:  Eigenvalues are values λ such that Av = λv, so you find them by solving Av − λv = 0, or equivalently, (A − λI)v = 0, where I is the identity matrix, or equivalently, |A − λI| = 0. That last form is also known as the characteristic equation of A, and, being a polynomial of the same degree as the dimension of A, you can find its solutions, and thus the eigenvalues of A, using a polynomial root finder.

Thomas is fully right.

You can find a program to compute the coefficients of the Characteristic Polynomial (its roots are the eigenvalues) for real or complex matrices in my article:

HP Article VA047 - Boldly Going - Eigenvalues and Friends

Also, there are many solved examples in the article, to make the matter crystal-clear.

V.
Well, a few things should still be mentioned here:
1. for the characteristic polynomial p(x)=|A-xI|, the term |A-xI| denotes the determinant of (A-xI).
2. the algebraic multiplicity of an eigenvalue (i.e. the multiplicity as zero of p) is always >= geometric multiplicity (i.e. the dimension of the eigenspace belonging to the eigenvalue).
3. I am not sure if "Dimension of a Matrix" is a valid definition. It's simply the number of rows (or columns).
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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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