A simple observation on random matrices with continuous diagonal entries

Let $T$ be an $n\times n$ random matrix, such that each diagonal entry $T_{i,i}$ is a continuous random variable, independent from all the other entries of $T$. Then for every $n\times n$ matrix $A$ and every $t\ge0$ $$ \p\Big[|\det(A+T)|^{1/n}\le t\Big]\le2bnt, $$ where $b>0$ is a uniform upper bound on the densities of $T_{i,i}$.


introduction
In this note we are interested in the following question: Given an n × n random matrix T , what is the probability that T is invertible, or at least "close" to being invertible?One natural way to measure this property is to estimate the following small ball probability P s n (T ) ≤ t , where s n (T ) is the smallest singular value of T , In the case when the entries of T are i.i.d random variables with appropriate moment assumption, the problem was studied in [3,11,12,15,17].We also refer the reader to the survey [10].In particular, in [12] it is shown that if the entries of T are i.i.d subgaussian random variables, then where c, C depend on the moment of the entries.
Several cases of dependent entries have also been studied.A bound similar to (1.1) for the case when the rows are independent log-concave random vectors was obtained in [1,2].Another case of dependent entries is when the matrix is symmetric, which was studied in [5,6,7,8,9,19].In particular, in [5] it is shown that if the above diagonal entries of T Date: February 21, 2013.2010 Mathematics Subject Classification.60B20,15B52.
are continuous and satisfy certain regularity conditions, namely that the entries are i.i.d subgaussian and satisfy certain smoothness conditions, then The regularity assumptions were completely removed in [6] at the cost of a n 5/2 (independence of the entries in the non-symmetric part is still needed).On the other hand, in the discrete case, the result of [19] shows that if T is, say, symmetric whose above diagonal entries are i.i.d Bernoulli random variables, then where c is an absolute constant.
A more general case is the so called Smooth Analysis of random matrices, where now we replace the matrix T by A + T , where A being an arbitrary deterministic matrix.The first result in this direction can be found in [13], where it is shown that if T is a random matrix with i.i.d standard normal entries, then Further development in this direction can be found in [18], where estimates similar to (1.2) are given in the case when T is a Bernoulli random matrix, and in [6,8,9], where T is symmetric.
An alternative way to measure the invertibility of a random matrix T is to estimate det(T ), which was studied in [4,14,16] (when the entries are discrete distributions).Here we show that if the diagonal entries are independent continuous random variables, we can easily get a small ball estimate for det(A + T ), where A being an arbitrary deterministic matrix.
Theorem 1.1.Let T be an n × n random matrix, such that each diagonal entry T i,i is a continuos random variable, independent from all the other entries of T .Then for every n×n matrix A and every t ≥ 0 where b > 0 is a uniform upper bound on the densities of T i,i .
We remark that the proof works if we replace the determinant by the permanent of the matrix (see [4] for the difference between the notions).Now, we use Theorem 1.1 to get a small ball estimate on the norm and smallest singular value of a random matrix.
Corollary 1.2.Let T be a random matrix as in Theorem 1.1.Then and Corollary 1.2 can be applied to the case when the random matrix T is symmetric, under very weak assumptions on the distributions and the moments of the entries and under no independence assumptions on the above diagonal entries.Note that in this case when T is symmetric, we have Thus, in this case we get a far better small ball estimate for the norm Finally, in Section 3 we show that in the case of 2 × 2 matrices, we use an ad-hoc argument to obtain a better bound than the one obtained in Theorem 1.1.We do not know what is the right order when the dimension is higher.

Proof of Theorem 1.1
Before we give the proof of Theorem 1.1, we fix some notation.First, let M = A + T , and let M k be the matrix M after erasing the last n − k rows and last n − k columns.Also, let Ω k be the σ-algebra generated by the entries of M k except M k,k .
Proof of Theorem 1.1.We have where f k is measurable with respect to Ω k .We also have where the last inequality follows from the fact for a continuous random variable X we always have where b > 0 is an upper bound on the density of X.
Thus, we get Also, note that Therefore, Choosing ε j = t j/n , the result follows.
Corollary 1.2 now follows immediately.

The case of 2 × 2 matrices
As discussed in the introduction, we show that for 2 × 2 matrices the small ball estimate on the determinant obtained in Theorem 1.1 is not sharp.To do that, we use the well known fact that if X and Y are continuous random variables with joint density function f X,Y (•, •) then X • Y has a density function which is given by where f X , f Y are the density functions of X, Y , respectively.
We thus have the following.
Proposition 3.1.Assume that X and Y are independent continuous random variables, with Since X and Y are independent, f X,Y (x, y) = f X (x) • f Y (y).We estimate each term of (3.1) separately.Assume first that |z| ≤ For the first term, we have And, for the second, by (3.4) Plugging (3.6) and (3.7) into (3.5), the result follows.
Using Proposition 3.1, we immediately obtain the following: Corollary 3.1.Let X and Y be independent continuous random variables.Then for every t ∈ (0, 1) and every γ ∈ R, where b > 0 is a uniform upper bound on their densities.
Proof.Note that the function We also obtain the following corollary.
Corollary 3.2.Let T = {T i,j } i,j≤2 be a random matrix such that T 1,1 and T 2,2 are continuous random variables, each independent of all the other entries of T .Then for every t ∈ (0, 1) where b > 0 is a uniform upper bound on the densities of T 1,1 , T 2,2 .
Proof.We have,