SWhat changed
Frobenius extends positive-matrix spectral theory to finite irreducible nonnegative matrices. The positive spectral radius is a simple eigenvalue with strictly positive left and right eigenvectors, while all eigenvalues on its spectral circle are simple and are the radius times a full cycle of roots of unity. The primitive case is characterized by a positive matrix power; sufficiently high powers are then positive. The same paper develops cyclic block structure and reduction into irreducible parts.
Scientific scope
Frobenius extends positive-matrix spectral theory to finite irreducible nonnegative matrices. The positive spectral radius is a simple eigenvalue with strictly positive left and right eigenvectors, while all eigenvalues on its spectral circle are simple and are the radius times a full cycle of roots of unity. The primitive case is characterized by a positive matrix power; sufficiently high powers are then positive. The same paper develops cyclic block structure and reduction into irreducible parts. Finite real entrywise-nonnegative matrices, not arbitrary signed matrices or infinite-dimensional positive operators. The positive-radius statement concerns nontrivial irreducible matrices, excluding the exceptional1×1zero matrix. Irreducibility alone does not make the positive eigenvalue uniquely largest in modulus: imprimitive matrices have a peripheral cycle. Strictly positive eigenvectors and simplicity cannot be transferred unchanged to arbitrary reducible matrices. This is the 1912 extension, not sole invention of all Perron–Frobenius antecedents or later stochastic-mixing bounds.
Evidence
The linked original works support the scoped statement and contribution attribution. WRS has not independently replayed experiments or reconstructed a full mathematical proof. Finite real entrywise-nonnegative matrices, not arbitrary signed matrices or infinite-dimensional positive operators. The positive-radius statement concerns nontrivial irreducible matrices, excluding the exceptional1×1zero matrix. Irreducibility alone does not make the positive eigenvalue uniquely largest in modulus: imprimitive matrices have a peripheral cycle. Strictly positive eigenvectors and simplicity cannot be transferred unchanged to arbitrary reducible matrices. This is the 1912 extension, not sole invention of all Perron–Frobenius antecedents or later stochastic-mixing bounds.
Contributors
- The named authors in the research record · SCIENCE: Source-listed authors for the exact contribution; experimental subjects are not autonomous scientific authors. Source support and predecessor work retain separate roles. Individual effort shares and unresolved identities are not inferred.
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