Random matrices have simple spectrum

Abstract

Let M n =(ξ ij )1≤i,jn be a real symmetric random matrix in which the upper-triangular entries ξ ij , i < j and diagonal entries ξ ii are independent. We show that with probability tending to 1, M n has no repeated eigenvalues. As a corollary, we deduce that the Erdős-Rényi random graph has simple spectrum asymptotically almost surely, answering a question of Babai.

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Correspondence to Van Vu.

Additional information

T. Tao is supported by a Simons Investigator grant, the James and Carol Collins Chair, the Mathematical Analysis & Application Research Fund Endowment, and by NSF grant DMS-1266164.

V. Vu is supported by NSF grant DMS 1307797 and AFORS grant FA9550-12-1-0083.

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Tao, T., Vu, V. Random matrices have simple spectrum. Combinatorica 37, 539–553 (2017). https://doi.org/10.1007/s00493-016-3363-4

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Mathematics Subject Classification (2000)

  • 05C80
  • 05C50
  • 60C99