In this section, we relate each of the scheduling criterion with the bounds on V-BLAST capacity. The best criterion technique is the one that is able to provide tighter bounds on
where is the minimum i th diagonal value of of the n th layer of the k th user. In this section, we will find bounds that are independent of n and i for each scheduling technique.
Choosing a user with maximum minimum singular value of H
is the same as choosing a user with maximum minimum eigenvalue of . We have
This means that we are looking for the minimum i th diagonal value of . We can write it in the form of the following inequality:
is a squared Hermetian matrix, it can be written as
where U is a unitary matrix with orthonormal eigenvectors and Λk,n is the diagonal matrix of eigenvalues . Another approach is to use QR decomposition as done in .
Using simple matrix manipulations, the inverse is written as
The diagonal elements of are . Therefore,
For the sake of simplicity, we will henceforth refer to and as and , respectively. We now have
According to the inclusion principle for matrices, the minimum value of λmin occurs at n=1 and the minimum vale of λmax occurs at n=MT.
Now we have an upper bound given by
or by taking the minimum over K users
We will use this inequality to establish lower bounds on MaxMinSV.
For analytical tractability, we will focus on the case when MT=MR. The probability density function (pdf) of the smallest eigenvalue for the case of MT=MR is given by 
In this case, and , where μ and σ2 represent the mean and variance. From chapter 10 of , we see that the maximum of λmin will scale as . Therefore, we have a new bound on the capacity of VBLAST as
In this case, we start with the inequality
Here κ(k,n) is the condition number of the matrix defined as λmax/λmin. Now
The condition number κ is maximum when n=1, a lemma which follows from the inclusion principle of matrices. Therefore, we have
The pdf of s, the condition number of H
for the case of MT=MR, is given by 
As κ=s2, we can show using a variable transformation argument that
Now we find the upper bound on mink κ(k). Gordon et al.  provide bounds on inequalities for the minimum of random variables. These inequalities exist if a random variable ξ satisfies the (α, β) condition defined as
where α>0 and β>0.
Then, the random variable ξ is bounded as
Using the pdf of κ, we find that . Now the upper bound on the minimum is given by
Now we have
We start with the established inequality
Again from the inclusion principle, the minimum of the trace occurs at n=MT.
Since we know the last layer, the pdf of for the last n becomes conditional. If we solve for the unordered VBLAST algorithm, we will get
is a row vector containing MR i.i.d complex Gaussian entries. In this case, the trace will be a chi-squared random variable with 2MR degrees of freedom. The pdf in this case is given as
Now we use an upper bound on the maximum of the trace. Bertsimas et al.  provide us with a tight upper bound:
where μ and σ2 are mean and variances, respectively. In this case, μ=2MR and σ2=4MR. Now we have
Here we define an upper bound on capacity: