  The theory of compressed sensing tells us that recovering all k-sparse
signals requires a sensing matrix to satisfy that its spark is greater than 2k,
or its order of the null space property (NSP) or the restricted isometry
property (RIP) is 2k or above. If we perform elementary row or column
operations on the sensing matrix, what are the changes of its spark, NSP order
and RIP order? In this paper, we study this problem and discover that these
three quantitative indexes of sensing matrices all possess invariance under all
matrix elementary transformations except column-addition ones. Putting this
result in form of matrix products, we get the types of matrices which multiply
a sensing matrix and make the products still have the same properties of sparse
recovery as the sensing matrix. According to these types of matrices, we made
an interesting discovery that sensing matrices with deterministic constructions
do not possess the property universality which belongs to sensing matrices with
random constructions.
