A generalization of Cauchy-Khinchin-van Dam inequality

We first give an alternative proof of a theorem originally presented by E. R. van Dam. Then we show a generalization of the van Dam matrix inequality.


Introduction
gave an upper bound on the sum of squares of degrees in a graph by considering some positive semidefinite quadratic form related to the line graph of the complete graph. Following de Caen's idea, van Dam (1998) gave a matrix inequality, which generalizes the Cauchy-Schwarz inequality for vectors, and Khinchin's inequality for zero-one matrices. In Section 2, we first present a different proof of Theorem 1 of van Dam (1998). In Section 3, we give the main result of this paper, a generalization of the van Dam matrix inequality. Then, in Section 4, we compare with the result of Yan (2011). Theorem 1 (van Dam, 1998, Theorem 1 PUBLIC INTEREST STATEMENT D. de Caen had gave an upper bound on the sum of squares of degrees in a graph by considering some positive semidefinite quadratic form related to the line graph of the complete graph. Following de Caens idea, E. R. van Dam gave a matrix inequality, which generalizes the Cauchy-Schwarz inequality for vectors, and Khinchins inequality for zero-one matrices. In this paper, we present a different proof of van Dam's inequality and then give a generalization. Finally, we compare with the generalization of Zizong Yan (2011). We hope that the result can be used to the investigation of quantum entanglement.

An alternative proof of van Dam's theorem
The equality holds if and only if a ij = b i + c j for some real b i and c j , i = 1, 2, … , m, j = 1, 2, … , n.
van Dam (cf. 1998) proved the theorem using the positivity of the matrix nI n − J n , where I n is the identity matrix of order n and J n is the square matrix of order n with all elements are equal to 1.
It is easy to see that the matrix nI n − J n has eigenvalue 0 with multiplicity 1 and n with multiplicity n − 1. By n = 1

Main result
We fix some notations which will be used in the following: The equality holds if and only if Vec(A) is in the kernel of (n 1 I n 1 − J n 1 ) ⊗ (n 2 I n 2 − J n 2 ) ⊗ ⋯ ⊗ (n s I n s − J n s ). Here, a i 1 i 2 …i s is the i 1 (n 2 … n s − 1) + i 2 (n 3 … n s − 1) + ⋯ + i s−1 (n s − 1) + i s component of Vec(A).

Notation:
We have an orthonormal basis for the eigenspace with eigenvalue 0 (i.e. kernel) of For example, when s = 3, we have where n 1 = m, n 2 = n, n 3 = l.

The case of s = 3
For the case of s = 3, Yan (2011) present another inequality.
Theorem 3 (Yan, 2011, Theorem 1.1) Let A = (a ijk ) be a real m × n × l and , , are real numbers. Then Next, we will have a comparison between these two results.