We prove an optimal estimate of the smallest singular value of a random subgaussian matrix, valid for all dimensions. For an N × n matrix A with independent and identically distributed subgaussian entries, the smallest singular value of A is at least of the order √ N − √ n − 1 with high probability. A sharp estimate on the probability is also obtained.
The following optimal estimate of the smallest singular value of the analysis matrix for a random subgaussian frame with independent entries is due to Rudelson and Vershynin REF .
Smallest singular value of a random rectangular matrix
Comm. Pure Appl. Math
Mathematics