Durrant and ata kabanwhere mkd random matrix with entries rij

A COMPARISON OF THE MOMENTS OF A QUADRATIC

FORM INVOLVING ORTHONORMALISED AND NORMALISED

In our recent papers [3, 4, 2] we asserted without proof that two particular exponentiated quadratic forms were (in each respective case) bounded above by the moment generating function of a chi-squared distribution. This report fills that gap by proving that those quadratic forms satisfy their corresponding inequalities, and that moreover such a bound holds for each moment individually.

2. Results

3. Proofs

3.1. Proof of Theorem 1. We want to show that: ER [( vTRT ( RRT )−1Rv )i]⩽ ER [( 1 σ2d· vT RT Rv	)i]
	)i]	(3.1)

values of RTR are the eigenvalues of RRT. Furthermore, since RRT∈ Mk×k, RRTis invertible. It now follows that if xj is an eigenvector of RTR with non-zero eigenvector λ(xj), then Rxj is an eigenvector of
vTRT ( RRT )−1Rv		=	∑			(3.3)
vTRT ( RRT )−1Rv		=	∑	=	{j:λ(xj)̸=0}∑	(3.3)
1Since	( RRT )−1/2RRT (( RRT )−1/2)T	=	( RRT )−1/2 ( RRT )1/2 (

MOMENTS OF RP MATRICES 3

We can now rewrite the inequality (3.1) to be proved as the following equivalent problem. For all i ∈ N:

Now, in RHS of (3.4) the αj depend only on v and the xj and these are both inde-pendent of λ(xj) (e.g. [1] Proposition 4.18, [5] Lemma 2.6) and so α is independent of λ. Hence we can rewrite this term as:

Eα		{j:λ(xj)̸=0}∑		i		(3.5)

ER [λ(xj)] = 1 k {j:λ(xj)̸=0}∑ ER [λ(xj)] = 1 kER [ Tr ( RRT )] = 1 k∑ER [ rT jrj ] (3.7)

where rj is the j-th row of R.

3.2. Proof of Corollary 1. To prove the corollary we rewrite the inequality (3.1)

using the Taylor series expansion for exp to see that:

References

[1] M. Artin. Algebra. Pearson Education, 2010.