Geodesic
Linear Transformations on Matrices: the Invariance of the Third Elementary Symmetric Function
Canadian journal of mathematics, Tome 22 (1970) no. 4, pp. 746-752

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Let T be a linear transformation on Mn the set of all n × n matrices over the field of complex numbers, . Let A ∈ Mn have eigenvalues λ1, ..., λn and let Er (A) denote the rth elementary symmetric function of the eigenvalues of A : Equivalently, Er (A) is the sum of all the principal r × r subdeterminants of A. T is said to preserve Er if Er [T(A)] = Er (A) for all A ∈ Mn . Marcus and Purves [3, Theorem 3.1] showed that for r ≧ 4, if T preserves Er then T is essentially a similarity transformation; that is, either T: A → UAV for all A ∈ Mn or T: A → UAtV for all A ∈ Mn, where UV = eiθIn, rθ ≡ 0 (mod 2π). 
Beasley, Leroy B. Linear Transformations on Matrices: the Invariance of the Third Elementary Symmetric Function. Canadian journal of mathematics, Tome 22 (1970) no. 4, pp. 746-752. doi: 10.4153/CJM-1970-084-x
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[1] 1. Marcus, M. and Mine, H., A survey of matrix theory and matrix inequalities (Allyn and Bacon, Boston, 1964). Google Scholar

[2] 2. Marcus, M. and Moyls, B. N., Transformations on tensor product spaces, Pacific J. Math. 9 (1959), 1215–1221. Google Scholar

[3] 3. Marcus, M. and Purves, R., Linear transformations on algebras of matrices: The invariance of the elementary symmetric functions, Can. J. Math. 11 (1959), 383–396. Google Scholar

[4] 4. Marcus, M. and Westwick, R., Linear maps on skew-symmetric matrices: The invariance of the elementary symmetric functions, Pacific J. Math. 10 (1960), 917–924. Google Scholar

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