Linear Algebra : Left & Right Eigenvectors and Eigenvalues

Let A matrix, vector x, λ eigenvalue

1. Right Eigenvectors and eigenvalues:

A.x = λx

Example:

A = \begin{pmatrix} 5 & -7 &  & 7 \\ 4 & -3 & & 4\\ 4 & -1 & & 2 \end{pmatrix} and, x = \begin{pmatrix} 1\\ 1\\ 1 \end{pmatrix}

Then,
A.x = \begin{pmatrix} 5\\ 5\\ 5 \end{pmatrix} = 5. \begin{pmatrix} 1\\ 1\\ 1 \end{pmatrix} = 5.x
So,
(right) eigenvalue = 5
image

2. Left Eigenvectors and eigenvalues:

x.A = λx

However, be careful that:
x.A = \begin{pmatrix} 1 & 1 && 1 \end{pmatrix}.\begin{pmatrix} 5 & -7 &  & 7 \\ 4 & -3 & & 4\\ 4 & -1 & & 2 \end{pmatrix} = \begin{pmatrix} 13 & -11 && 13 \end{pmatrix}

If we want to find the left eigenvector associated with the eigenvalue 5, then we find the eigenvector A^T.
(https://en.m.wikipedia.org/wiki/Eigenvalues_and_eigenvectors#Applications)

This would lead us to see that:
(-1 1 -1).A = (-5 5 -5) = 5. (-1 1 -1)
So, in this example, the eigenvalue 5 has different left and right eigenvectors:
(-1 1 -1) & (1 1 1) respectively.

Remark 1: However, the nice fact about matrices is that always :
left eigenvalue = right eigenvalue.
So we just simply call eigenvalue for short.

Remark 2: Intuitive explanation of eigenvector and eigenvalue.

Ref:
《Math Bytes》by Tim Charter
Princeton University Press
[NLB #510 CHA]

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