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How many eigenvalues of a random symmetric tensor are real? (English) Zbl 1431.15023
A. Edelman et al. [J. Am. Math. Soc. 7, No. 1, 247–267 (1994; Zbl 0790.15017)] computed the expected number of real eigenvalues of a matrix filled with i.i.d. standard Gaussian random variables. This model of random matrix is extended to matrices of higher order, called tensors. The author answers a question posed by J. Draisma and E. HorobeŇ£ [Linear Multilinear Algebra 64, No. 12, 2498–2518 (2016; Zbl 1358.15015)], who asked for a closed formula for the expected number of real eigenvalues of a random real symmetric tensor drawn from the Gaussian distribution relative to the Bombieri norm. This expected value is equal to the expected number of real critical points on the unit sphere of a Kostlan polynomial. The author also derives an exact formula for the expected absolute value of the determinant of a matrix from the Gaussian orthogonal ensemble.
MSC:
15B52 Random matrices (algebraic aspects)
15A18 Eigenvalues, singular values, and eigenvectors
60B20 Random matrices (probabilistic aspects)
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