A square matrix with strictly positive entries has a unique largest real eigenvalue, with a corresponding eigenvector that can be chosen to have strictly positive entries. Named for Oskar Perron and Ferdinand Georg Frobenius, it underlies applications from Markov chains to network ranking algorithms.
Facts
StatementA real square matrix with strictly positive entries has a unique largest eigenvalue in absolute value, and that eigenvalue is real and positive, with a corresponding eigenvector that can be chosen strictly positive. 2 Proof YearPerron proved the original strictly-positive-matrix case in 1907; Frobenius extended it to certain non-negative matrices in 1912. Classification
Statement Form Connections
Sources
1. Wikipedia: Perron-Frobenius theorem
WikipediaLead section, statement-form referenceQuote, Lead section, statement-form reference
In matrix theory, the Perron-Frobenius theorem, proved in its first part by Oskar Perron (1907) and extended by Georg Frobenius (1912), asserts that a real square matrix with positive entries has a unique eigenvalue of largest magnitude and that eigenvalue is real.
2. Perron-Frobenius Theorem (Wikipedia)
Wikimedia Foundationlead section
a real square matrix with positive entries has a unique eigenvalue of largest magnitude and that eigenvalue is real.
History section
Early results were due to Oskar Perron (1907) and concerned positive matrices. Later, Georg Frobenius (1912) found their extension to certain classes of non-negative matrices.
View the Source Reader Challenges (0)
No disputes yet. Spotted an error or a better source? Open the first one.
Sign in to dispute this or suggest a correction.