In this paper, we mainly study the smallest singular value of submatrices consisting of row vectors bounded by 1, and we establish that the minimal smallest singular value of submatrices of matrices of size n + 1 times n consisting of row vectors bounded by 1 is equal to 1Moreover, we establish that the minimal smallest singular value of submatrices of matrices of size n times 2 consisting of row vectors bounded by 1 is sharply bounded above by 2sin 2n , and furthermore, this bound is achieved if and only if the rows of diag ( 1 , 2 , , n ) A are the coordinates of n adjacent vertices of a regular 2n -gon on the unit circle S 1 in R 2 for someAdditionally, we show that the equiangular frames in the projective spaces do not form the matrices in the general dimensions with the optimal smallest singular value of the submatrices, contrary to the case of matrices of dimension n + 1 by n or negative to the conjectures based on the phenomena in the low dimensions.
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Yang Liu (2024) studied this question.
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