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We now give a simple inductive proof of Mirsky’s Theorem. Let [TΛ] denote the N-square matrix that has Λ as its diagonal sequence, and has 1 in every entry immediately above the diagonal, and 0 in all remaining entries. A unit lower triangular matrix, is a square matrix in which every entry on the diagonal is 1, and every entry above the diagonal is 0. The unit lower triangular matrices are a group under matrix multiplication. The following result includes Mirsky’s Theorem and a little more. Theorem 1.2. There exists a matrix A with eigenvalue sequence Λ and diago- nal sequence D if and only if condition (1.2) is satisfied. Indeed, under this condi- tion, there exists a unit lower triangular matrix L such that the diagonal sequence of −1 L [TΛ]L is D. If Λ and D are both real, L can be taken to be real. Proof: The necessity of (1.2) for the existence of A is obvious, and we must prove that (1.2) is sufficient for the existence of L. For N = 2, direct computation shows that, 1 0 with L = , the diagonal sequence of L−1[T ]L is (λ + c, λ − c). Choosing  c 1  Λ 1 2 c = d1 − λ1 yields the assertion for N = 2. Assume that the assertion is proved for K-square matrices of size up through K = N − 1. Given N-sequences Λ and D satisfying (1.2), let L be the 2-square unit −1 lower triangular matrix such that L [T(λ1,λ2)]L has the diagonal sequence (d1, λ2), where e (1.3) λ2 = λ1 + λ2 − d1 . Let L be the N-square matrix obtainede by replacing the upper left 2-square block − of I × by L. Then L 1[T ]L has the following properties: Its upper left 2-square Ne N Λ block has diagonal sequence (d , λ ), its lower right (N − 2)-square block is the same e e1 2 −1 as that of [TΛ], and L [TΛ]L = 1. 2e,3 h i − 1 e The lower right (Ne − 1)-squaree block of L [TΛ]L is [TΛ] where Λ is the (N − 1)- sequence (λ2, λ3,...,λN ). By the inductive hypothesis, there exists a unit lower −1 e e e triangular matrix M such that M [Te]M has the diagonal sequence (d ,...,d ). e Λ 2 N Now let M be the N-square matrix obtained by replacing the lower right (N − 1) −1 I × M LM T LM D square blockf of N N with . Then ( ) [ Λ]( ) has the diagonal sequence , and LM is unit lower triangular. e f e f Our proof of Theorem 1.2 is related to a proof of Horn’s Theorem by Chan and e f Li [2]. For the reader’s convenience, we briefly recall their proof in our notation, and then conclude with a brief comparison of the proofs. Proof of Horn’s Theorem, after Chan and Li: Assume that Λ and D are in decreasing order. If N = 2, then Λ ≻ D implies λ1 ≥ d1 ≥ d2 ≥ λ2. For λ1 > λ2, 1/2 1/2 −1/2 (d1 − λ2) −(λ1 − d1) U = (λ1 − λ2) 1/2 1/2  (λ1 − d1) (d1 − λ2)  ∗ is a real orthogonal matrix, and the diagonal sequence of U [Λ]U is D = (d1, d2). For λ1 = λ2,Λ= D, and we may take U = I. This proves Horn’s Theorem for N = 2. We proceed inductively: Let Λ and D be two real N-sequences with Λ ≻ D. There is a K with 1 ≤ K identity matrix IN×N , and replace the 2-square diagonal block at positions K,K +1 by the 2-square matrix U. Then T ∗[Λ]T has the diagonal sequence Λ′, which ′ is obtained from Λ by replacing λK by dK and λK+1 by λK+1 in Λ. We note the important fact that Λ′ ≻ D. What follows is especially simple if K = 1, and the reader may wish to consider that case first. Let Λ′′ and D′′ be the sequences of N − 1 real numbers obtained ′ ′′ ′′ by deleting dK , the common Kth term in both Λ and D. Then Λ ≻ D , and hence, by induction, there exists an (N − 1)-square orthogonal matrix V such that V [Λ′′]V ∗ has the diagonal sequence D′′. It is convenient to index the entries of V using {...,K − 1,K +1,...} leaving out the “deleted” index K. Promote V to an N-square orthogonal matrix by setting VK,K = 1 and Vj,K = VK,j = 0 for j 6= K. Then (VT )[Λ](VT )∗ has D as its diagonal sequence. While Chan and Li’s proof of Horn’s Theorem and our proof of Mirsky’s Theorem have a similar structure, there is a significant difference: The matrices whose existence is guaranteed by Horn’s Theorem are all necessarily similar to the diagonal matrix [Λ], so that [Λ] provides a suitable starting point for the proof. This is not the case for Mirsky’s Theorem: Indeed, if all entries of Λ are equal, [Λ] is a multiple of the identity. Thus, the set of all matrices similar to [Λ] contains only [Λ] itself. Except in the case D = Λ, it does not contain any matrices with diagonal sequence D. However, our proof shows that in the set of all matrices similar to [TΛ], there is always one with diagonal sequence D provided that (1.2) is satisfied. Acknowledgement: We thank Prof. Roger Horn for invaluable advice.

REFERENCES

[1] G. Birkhoff. Tres obsevaciones sobre el algebra lineal. Univ. Nac. Tucum´an Rev. Ser. A 5, 147–151,1946. [2] N.N. Chan and K.H. Li. Diagonal elements and eigenvalues of a real symmetric matrix J. Math. Analysis Appl. 91, 561-566, 1983. [3] A. Horn. Doubly stochastic matrices and the diagonal of a rotation matrix Amer. J. Math. 76, 620–630, 1954. [4] L. Mirsky. Matrices with prescribed characteristic roots and diagonal elements J. London Math. Soc. 33, 14–21, 1958. [5] I. Schur. Uber¨ eine Klasse von Mittlebildungen mit Anwendungen auf der Determinantentheorie Sitzungsber. Berliner Mat. Ges. 22, 9–29,1923.