Concerning diagonal similarity of irreducible matrices
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- by D. J. Hartfiel PDF
- Proc. Amer. Math. Soc. 30 (1971), 419-425 Request permission
Abstract:
If $A = ({a_{ij}})$ is an $n \times n$ irreducible matrix, then there are positive numbers ${d_1},{d_2}, \cdots$, ${d_n}$ so that $\sum \nolimits _k {{d_i}{a_{ik}}d_k^{ - 1} = } \sum \nolimits _k {{d_k}{a_{ki}}d_i^{ - 1}}$ for each $i \in \{ 1,2, \cdots ,n\}$. Further, the numbers ${d_1},{d_2}, \cdots$, ${d_n}$ are unique up to scalar multiples.References
- Richard A. Brualdi, Seymour V. Parter, and Hans Schneider, The diagonal equivalence of a nonnegative matrix to a stochastic matrix, J. Math. Anal. Appl. 16 (1966), 31–50. MR 206019, DOI 10.1016/0022-247X(66)90184-3 M. Fiedler, Bounds for eigenvalues of doubly stochastic matrices (submitted).
- F. R. Gantmaher, Teoriya matric, Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow, 1953 (Russian). MR 0065520
- Richard Sinkhorn and Paul Knopp, Concerning nonnegative matrices and doubly stochastic matrices, Pacific J. Math. 21 (1967), 343–348. MR 210731
Additional Information
- © Copyright 1971 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 30 (1971), 419-425
- MSC: Primary 15.60
- DOI: https://doi.org/10.1090/S0002-9939-1971-0281731-5
- MathSciNet review: 0281731