BrainHurts said:Homework Statement
If A≥0 and Ak>0 for some k≥1, show that A has a positive eigenvector.
Homework Equations
The Attempt at a Solution
A is nxn
Well from a previous problem we know that the spectral radius ρ(A)>0
We also know that if A≥0, then ρ(A) is an eigenvalue of A and there is a non negative vector x, x=/=0 such that Ax=ρ(A)x
Kinda stuck
Non-negative matrices are matrices where all elements are equal to or greater than zero. In other words, there are no negative numbers in the matrix.
Non-negative matrices have a wide range of applications in fields such as computer science, economics, biology, and social sciences. They are often used to model and analyze complex systems, such as transportation networks, social networks, and biological processes.
The main difference between non-negative matrices and regular matrices is the restriction on the values of their elements. While regular matrices can have positive, negative, and zero elements, non-negative matrices can only have elements that are equal to or greater than zero.
No, non-negative matrices cannot have negative eigenvalues. This is because the eigenvalues of a non-negative matrix must be equal to or greater than zero, since the elements of the matrix are also non-negative.
Non-negative matrices are useful in data analysis as they can be used for dimensionality reduction and data clustering. They can also be used in the process of data decomposition, where a larger dataset is broken down into smaller, non-negative matrices for easier analysis and interpretation.