Lagmental Vicfred

Perron--Frobenius Gives a Positive Dominant Eigenvector by Vicfred

tags: mathematics, matrix factorizations, least squares, positive matrices

A positive matrix has a simple spectral-radius eigenvalue with positive left and right eigenvectors. Keeping the exact identity in view prevents the geometric or probabilistic intuition from drifting.

The mathematical object

Matrix factorizations expose different geometry: \(A=QR\) separates an orthonormal frame, while \(A=U\Sigma V^\ast\) separates rotations from axis scaling.

$$ A_{ij}>0 $$

I keep the defining relation \(\mathsf D\) above the derived relation \(\mathsf C\). This exposes whether cancellation used \(x\ne0\) and whether the conclusion is canonical.

$$ Ar=\rho(A)r,\qquad r_i>0,\quad|\lambda|<\rho(A)\ \text{for every other eigenvalue }\lambda $$

One explicit computation

The middle display is intentionally dense: it is where signs, bounds, multiplicities, or normalising factors are most likely to be lost.

$$ \min_i\frac{(Ax)_i}{x_i}\le\rho(A)\le\max_i\frac{(Ax)_i}{x_i}\qquad(x_i>0) $$

Why the identity matters

The aligned summary deliberately puts the datum and conclusion on different rows. Mathematically, this is the distinction between specifying an object and proving a property of it.

$$ \begin{aligned} \mathsf{D}\;&:\quad A_{ij}>0,\\[5pt] \mathsf{C}\;&:\quad Ar=\rho(A)r,\qquad r_i>0,\quad|\lambda|<\rho(A)\ \text{for every other eigenvalue }\lambda. \end{aligned} $$

Where it can fail

Conditioning matters numerically. An exact algebraic identity can be a poor computational method when it squares the condition number.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] Ar=\rho(A)r,\qquad r_i>0,\quad|\lambda|<\rho(A)\ \text{for every other eigenvalue }\lambda \end{gathered}} $$

The notation is dense, but it is doing honest work: every delimiter records scope and every index records dependence. Removing one should require a mathematical reason.

This article was posted on Wed 09 December 2020. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.